Good couch co-op/multiplayer games?

I had a blast playing Overcooked 1 & 2 with my partner and they run fine on Deck. I only played with 4 players once and it was a bit chaotic since it was the first time playing Overcooked for most of us.
If you plan on emulating Wii U, New Super Mario Bros. U sounds like a fun option. In the same vein, Rayman Origins and Legends are great co-op platformers too and are both Gold on ProtonDB

Overcooked (4-player)
Lovers in a dangerous space time (4-player)
Death squared (2-4 player)
Toybox Turbos (4-player)
Trine (3-player)
Keep Talking and Nobody Explodes (infinite player)
Magicka (4-player)
The various Jackbox Party Packs (players depends on which game. Most support 4 players, some up to 100 players, each player must have a phone to play on)
Worms W.M.D (6-player)
Untitled Goose Game (2-player)
- Love in a Dangerous Space Time is a great co-op game that is actually cooperative.

Shredder's revenge
It takes two
Gang beasts
Rivals of aether
Screen cheat
- Screen Cheat is so damn underrated. Love that game
  
  I know, right!? I haven't played it in a long time, but had tons of fun with it when I was. Great party game and especially great for people who remember what it was like playing split screen back in the day 😅

I always search steam sales for local multiplayer games. I have not tested all of these yet, so I'm going to categorize them here.
Games I already played with someone (e.g. "tested")
Boomerang Fu
Brawlhalla
Castle Crashers
Gang Beasts
Guacamelee - Super Turbo Championship Edition
Helldivers
Lovers in a Dangerous Spacetime
Regular Human Basketball
Just Shapes and Beats
Lethal League Blaze
MageQuit
Magicka / Magicka 2
Make Way
Overcooked
Road Redemption
Speedrunners
Towerfall Ascension
Tricky Towers
Ultimate Chicken Horse
Wobbly Life
Games for future play sessions (not yet tested)
Barony
Beat Me
Chained Together
Fling to the finish
Geometry Wars 3
Goat Simulator
Party Club
Pummel Party
Screencheat
Sonic Segal All Stars Racing
Stick Fight the Game
Treadnauts
Unrailed
Have fun :)
- avoid Magicka 1 for multiplayer if you can. The final update they pushed before abandoning it turned multiplayer into a buggy, crashy mess
  Edit: it might be fine for local couch multiplayer actually
  
  I don't plan on playing any of these games over any kind of network anyway. I'm all for couch coop / pvp, it just hits differently :)
- Dude...I knew/had only a few in this list. And most of the rest look amazing. This is a long list of very promising games, thanks so much!
  
  You're welcome, I'm glad to spread the old-school pre-internet local couch coop fun :)
  My personal favourites are
  MageQuit
  This is the most addicting of all the played games. I bought this with a "fun little magic-based pvp-only game for now and then" mindest. I thought "super smash brothers but magic". I started playing it with my friend on his TV "just for an hour" and suddenly, it was dark outside and time to go home.
  The next meeting we planned on playing MageQuit for a round or two and then move on to one of the other, yet unplayed, games. The moving on part never happened, MageQuit was just too much fun.
  Lovers in a Dangerous Spacetime
  This is the game for the whole family. You (up to 4 players) are in a spaceship. The spaceship has different buttons and levers in different places to control different things like acceleration, changing direction, aiming / firing weapon, directing partial shield or countermeasure etc. and you need to rescue your bunny-friends.
  They are scattered around the levels, sometimes hidden, sometimes locked up, sometimes guarded etc and you need to work together with your teammates to direct the spaceship. You get quite a few different weapons and shields / countermeasures, which can also be combined, you upgrades for the ship, can buy different ships etc.
  It looks and sounds adorable, but if you don't work together, it's way harder then it looks. This is a game with a campaign and story.
  Regular Human Basketball
  Think basketball, but stupid and fun. The regular humans are actually motionless robots which need to be moved by using switches and levers inside it, which is what your job is. You even have a jet-boost at some parts of your regular-human body. We laughed our asses off.
  It is similar to Lovers in a Dangerous Spacetime in the sense that, you need to work together to control a bigger machine. This is just a pvp only game, no story or campaign.
  Ultimate Chicken Horse
  Race each other to the finish of an obstacle course. After each round, everyone picks a new obstacle to place and expands the course. Seldomly have I ever seen such bullshittery as my friends and me created in this game and then had to go through.
- I can confirm that Barony is a fun couch coop experience for two on a steam deck, so long as you are prepared to die and have to start over a lot. It might be tricky with four players, as there is a lot of inventory menuing. You might need a really large screen.
  
  Good to know, thank you!
  We always play on someones TV anyway. These are typically 46" ore bigger and have at least FullHD resolution. Would this be manageable?

Speedrunners
It's on sale for $3 right now. Awesome 4 player competitive racing game without split screen so everyone gets maximized view. Way different than Mario Kart so it won't be a repetitive experience for them. This is the Steam-based couch coop that's lasted longest in my friends group.
Tricky Towers
Tetris game, but the catch is you build upwards to a finish line and physics play a role, so the tower can lean and collapse.
TowerFall Ascension
Competitive platformer shooter. No split screen. One large screen for everyone. My favorite aspect is all the options you can change to switch up the gameplay. Coop mode and PvP mode.
Mount Your Friends
A personal favorite of mine. Not necessarily for kids, even though I don't think it's anything beyond maybe PG-13. It's hilarious. It can be pretty competitive. You can choose to build upwards or outwards. I think outwards is more fun. You can watch gameplay videos and determine if it's appropriate for the kids in question.
Others have pointed out games like A Way Out, the Trine series, and Split Fiction. Those are all great. But they're longer, story-based puzzle games that take more time than you may have for them to finish. Trine allows 3 players, but the others only allow 2.
- Mount Your Friends also has a 3D version (Mount Your Friends 3D) which is even more hilarious and adds new challenge given the full 3D-ness. 😃
- Can second SpeedRunners. They've even announced SpeedRunners 2 recently. The only downside with the game is that when you master the mechanics in the game you can't really play with your friends anymore, unless they are also keeping up. But even then I think I played the game many tens of hours with my friends before I got addicted and got ahead of them.

Towerfall
Gang beasts
Overcooked 2
Broforce
Or taking turns in 1v1 games

don't sleep on Portal 2
- Agreed, although it's wise to have a backup option for this. It's entirely possible that you have two solid Portal players go at this, which should be a really fun romp. However, in my experience, any skill gap between the players usually turns every stage into "how do I carry my friend through this puzzle?" An extreme version of this can be seen in Game Grumps' playthrough.

I was having fun with my 5 year old and It Takes Two, but the damned wasp boss level cut our adventure short. It proved too much for the kid
- I greatly enjoyed It Takes Two with my wife.
  Currently on split fiction by the same developers, it's amazing as well but a bit more challenging. A Way Out is also decent but we strongly disliked the end.
  Unraveled 2 was also very nice but a bit darker.
  
  That's interesting that you thought split fiction was harder than it takes two. My partner and I thought the opposite, since split fiction seems to let you get away with skipping a lot of things when one of you is dying (i.e. checkpoint reached by one player counts for both of you). In saying that my other theory is my partner has actually gotten better at games since we played it takes two.
- You might have saved him from a rather...traumatic experience right on the next chapter :)
  
  Welp... Good to know

Heave Ho
Stick Fight The Game
All Sonic Racing games (Transformed is the best one but sometimes gets confused with inputs, you may need to delete a config file once in a while but it's not a Steam Deck-specific problem)

Stardew Valley, a lot of Nintendo games using emulators(seriously just pick a game)

I had a lot of fun with TMNT: Shredders Revenge
It's a super fun sidescroller that sends you back to the good old SNES times.
You can play with up to four players.
The only downside, it's relatively short, especially for the price they ask. I think it's curently on sale though.

Overcooked 2, Conduct Together/Deluxe. Both are tough but fair, and they'll have to communicate effectively to do well.
Ultimate Chicken Horse is a bit more chill, more of a party game.

D&D Chronicles of Mystara (side-scrolling co-op) I think is 4-player
Cuphead - 2 player
A Hat In Time - 2 player
Streets of Rogue - up to 4 players
Vampire Survivors - I think 2 player for local co-op?
Most of the old Telltale games have co-op in the form of voting for responses
Binding of Isaac: Rebirth has full co-op now, I think up to 4 player. You may need the DLC for the best co-op experience
Crypt of The Necrodancer has solid couch co-op
Baldur's Gate 3 has a full split-screen 2-player couch co-op mode. Both players need to use a controller. Works great.
- I can’t help but laugh at the idea of a couple of 8 year-olds playing through Baldur’s Gate 3 lol.
  
  Lmao, that'll teach me not to skim ¯(ツ)/¯

I would recommend Overcooked and Moving out, the kids i played this with picked it up fairly quickly.
Another good coop game is Unrailed, but it is more stressful and might be more frustrating.
Our most played game is ultimate chicken horse though, but its versus and does not really provide guardrails so you can kind of deadlock your match quite easily

Moving Out and Overcooked are my favorite co-op games right now. Both have more than 1 title in the series, and, so far, none of them have been bad.

Cuphead.

Strong, STRONG recommendation for checking out Hypercharge: Unboxed.
It’s an amazing mix of aesthetic from Army Men: Sarge’s Heroes on the N64 and the Toy Story 2 Action Game with the PvP gameplay of the former and PvE gameplay of any famous shooter from the past 20 years.
To top it off, the level of detail put into every damn corner of the game is to die for. Customizable action figure player characters, all the NPC enemies are recognizable toys that react exactly as you expect (i.e. Beyblades that spin out into pieces when destroyed, and plastic soldiers that melt when shot with a flamethrower), and the environments are all lovingly designed and decorated. It’s the perfect modern couch-co-op game.
- Thanks a lot! How many split-screen players are allowed? Only two, or does it go up to 4?
  
  It supports up to 4. Full disclosure I haven’t tried split screen on the Deck, but I fully believe it’s worth a shot

River City Girls is free on Epic right now. Side-scroller fighting game a la Ninja Turtles. Only 2P though.

Been having a blast with Make Way lately, it's a top down racer where you start each round by building new sections of the track and adding traps to it. It's really chaotic and fun, and supports up to 6 local or online players
- Thanks! Second time I see it recommended, I'm definitely getting this one.

Check out Armed and Gelatinous on steam. Up to four players can battle each other. It’s an Indie game and it’s a lot of fun.

Streets of Rage 4
Edit: just saw that this is for playing with kids so maybe not a great one for that, sorry! I've had a blast with it playing with friends though.

Mount Your Friends is an Xbox 360 era go-to for me. Very easy to get into and gets people competitive (in a good way). The only downside is it's probably not the most appropriate game to play with kids for your situation

Assault Android Cactus!

If you like roguelites ember knights is awesome as a group of 4

A lesser known multiplayer game that's less co-op and more competitive fun is ultimate chicken horse. It's a super silly game where everyone compete platform style to reach the end of a stage, but each round everyone places another trap. It can get pretty funny pretty quick!
Recommend using the in game cheat to unlock all stages so there's more variety right away

One of my favorite in-person MP games of all time is Towerfall Ascension. My family and I used to spend so much time laughing out loud while playing that game. So many ridiculous trick shots with all the power ups. I haven't tried it on Steam Deck, but I don't see any reason why it wouldn't work, assuming you can get multiple controllers connected.

I found Brotato to be fun.
If you like 80's action movies, there is Broforce.
Duck Game is good as a platform shooter.

SpeedRunners, Ultimate Chicken Horse, Duck Game, Towerfall Ascension, and Stick Fight are some of my favorites. Crawl is also really good, but a bit more complex so depends on the age of the kids. Also can never go wrong with a platform fighter, Rivals of Aether 2 is great.