I recently had a terrible experience flying with them (delayed flights on a last-minute trip to a funeral over Labor Day weekend) but one thing I felt better about was that it was at least not Boeing.
Someone commented on another video that they saw the Ram Air Turbine extended. So they would've lost power, supporting your electrical fire theory. Also it seems extending the RAT disables some safeguards, that can cause the wheels to lock and catch fire.
From the picture, looks like a fire on one of the wheels. I'm guessing something about the brakes. The article described it as a "hard landing", so maybe something broke and jammed, then caught fire from friction.
Shit happens, especially with a lot of flights every day. Truck brakes catch fire sometimes too. This isn't particularly unusual.
Safe is relative. Air travel is safer than a rocket car, less safe than staying home. Generally the risk is acceptable, especially since the rate of injury per passenger-mile is lower than driving.
Just the front of the engine? Because that's what it looks like to me ... that the rear of the engine is still attached to the wing, but the front of it is sagging.