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.)
Cayuga is a connie according to memory alpha - confirmed in some background graphics: https://twitter.com/timothypeel1/status/1690018124492349440/photo/1