The problem with a lot of sci-fi is that if you considered what effects the technology would truly have on society you very quickly get a situation that is so different from the common experience of the audience that most can't easily get on board any more.
If the Federation allowed the full range of genetic engineering that we know they have the tech for, nobody would get old or die of natural causes. Gender and even species would be fluid. People could have arbitrary numbers of parents, or family relationships that the word "parent" doesn't meaningfully apply to. I would love to watch a show in a setting like this but I suspect that my taste wouldn't be so common. And the special effects and makeup budgets would skyrocket.
The Federation's ban on AI and GE also has tinges of authoritarianism that run counter to the liberal ideal it's supposed to represent. Say some planet in the Federation takes a different view of these issues, and wants to create a race of super-Datas and give them equal rights, are we expected to believe that the rest of the Federation will show up to bust down their doors, like some kind of space-DEA?
It's not just story-technical. It's also a connonical issue. The federation is, by al means, equal. Everyone is equal. Personal ambition can be achieved by merit.
If you'd introduce genetic engineering (GE) in the mix, you'd have a system where personal ambition can only be achieved through GE. People are forced to GE thrmselves, or the'll be left behind. The federation doesn't need currency, but GE could become the new currency where only those qith the right and enough GE get the best treatment.
Except the Federation is very much not equal. There's plenty of genetic variation within species and a much larger variation between species. Vulcans live hundreds of years and have telepathy, humans die after just one hundred and are brain-blind. Bashir was born with severe genetic defects. Geordi was blind, and then he could see better than baseline. There are differences in fundamental capabilities everywhere.
Genetic engineering would allow those advantages to be shared by everyone. It annoys me how hypocritical the Federation is about this issue.
I think its a reflection of the writers more than anything. Trek writers usually try to pull their ideas from current political and sociological issues. AI and genetic engineering (which I think is thinly veiled commentary on gender) are simply the current issues of our time. It is odd that they've chosen the federation to be the problem in the recent shows though. In older trek the federation was always the ones to stand up for the cause and the conflict came from worlds they visited. Of course, that's probably linked to the same idea as the AI/genetics, people are much more frustrated and disillusioned with western government these days and the writers are no doubt exploring that.
So there will probably be changes in the federations stance that are two steps beyond what we see happening in the world around us. Trek has always tried to lead people in a moral/philosophical direction.
Honestly the franchise is all over the place on this topic. Go back and watch S02E08 of ST:TNG, Unnatural Selection. No one getting arrested for genetic engineering there. My head canon is that by the time TNG is going genetic engineering has gone from truly taboo to just discouraged/not well accepted.