It's never made much sense that the entire multi-species Federation would be subject to a strict ban on genetic engineering due to events on Earth that happened centuries before the Federation was even founded. The way they doubled down on that rationale in Una's trial only highlighted the absurdity -- especially when Admiral April claimed he would exclude Una to prevent genocide.
On the one hand, the writers may be trying to create a straw man out of a weird part of Star Trek lore so they can have a civil rights issue in Starfleet. And that's fine. From an in-universe perspective, though, I think we can discern another reason for the ban on genetic engineering -- the Klingon Augment Virus.
There was a ban on genetic engineering on United Earth, which is understandable given that it was much closer to the time of the Eugenics Wars. Why would that remain unchanged when more time passed, more species joined, and more humans lived in places without living reminders of the war? [NOTE: I have updated the paragraph up to this point to reflect @Value Subtracted's correction in comments.] The answer is presumably that they needed to reassure the Klingons that something like the Augment Virus would never happen again. Hence they instituted a blanket ban around that time -- perhaps in 2155, the year after the Klingon Augment Virus crisis and also, according to Michael Burnham, the year the Geneva Protocols on Biological Weapons were updated.
That bought the Federation over a century of peace, but after war broke out due to a paranoid faction of Klingons who thought humans would dilute Klingon purity and after peace was only secured through the most improbable means, they doubled down on the ban. Una's revelation provided a perfect opportunity to signal to the Klingons that they were serious about the ban -- hence why they would add the charges of sedition, perhaps. Ultimately, an infinitely long speech and the prospect of losing one of their best captains combined to make them find a loophole -- but not to invalidate the ban or call it into question. This Klingon context is why April, who we know is caught up in war planning of various kinds, is so passionate that the ban exists "to prevent genocide" -- he's not thinking of people like Una, he's thinking of the near-genocide they suffered at the hands of the Klingons.
This theory still doesn't paint the Federation in a positive light, since they have effectively invented a false propaganda story to defend a policy that has led to demonstrable harm. But it makes a little more sense, at least to me. What do you think?
It makes sense that following the Eugenics War (and I believe it's heavily suggested that the Eugenics War and disagreements around genetic modification were a key cause of WW3) that genetic engineering would be banned on Earth. First Contact happens not long after WW3 in 2063. It takes almost 90 years for Enterprise to be launched in 2151 with the help of the Vulcans.
WW1 ended 105 years ago and it's still taught in schools, including the causes. WW2 ended 78 years ago and is also still heavily taught and is used as a parallel for why fascism is bad, even today. Both are a similar time difference from today as WW3 at the time of Enterprise and has a higher death toll (30% of the population) and so if you could ban a leading cause of that war, you would. I could see UE making it a requirement on the other founding members (none of which seems to have particularly experimented with it, and have seen the effect on Earth. Banning it is only logical.)
During that time, we see a rise up of the Soong's augments in 2154, which ends badly and I think would be more than enough to reinforce the prejudice in Earth society (and the Vulcans and Andorians would have been aware of it too.) Plus we see the Klingon Augment Virus as you suggested.
SNW takes place in 2260, which is about 100 years on and now the Federation has obviously been founded and is a growing force - but 100 years isn't that long and as human life times are suggested to be extending by then - there are probably people alive that remember the events of 2154. The rules have also led to fewer or no repeatable events involving augments and I can understand wanting to keep it that way.
Then in DS9, we see that it is still being practised illegally by some. Occasionally it turns out fine like Bashir but often leads to medical issues like Jack and his group in statistical probabilities. (Which again, ends with them trying to force the Federation to surrender to the Dominion.)