For one thing, I can't imagine a 24th Century Human Race that isn't composed of The Beautiful People. Nerds, geeks, misfits and outcasts are, more or less "bred out," if you like. Besides, what woman of the future would endure pregnancy and childbirth, when they can have test-tube babies that develop in vats of embryonic fluid for 9 months and are "born" once gestation is complete.
Even today ... would any parent really want their kid to suffer social awkwardness? Of course not. From the father's semen and the mother's eggs, there's almost too much choice involved, in custom-designing your future child. Thus helping to ensure that they'll grow up with every possible advantage. Unfortunately, this does invite "trends" and such. For example, "this year's trend" might be this or that complexion, or this or that hair texture ... and so on.
But more basic than that, even, strength of mind could be ensured, as well. Barclay's psychological impedimentia would be wiped out, along with ... hell, I don't know ... near-sightedness, or uh ... weight problems and so on. Not enhanced like genetic superheroes, or anything like that. Just normal Humans fully prepared to live Life to its fullest ... to achieve that "bettering" of oneself that the 24th Century holds so dear.
We explicitly know that this isn't the case at all in Trek, though; the only acceptable and legal genetic engineering in the Federation is to correct heritable diseases, and anything else is illegal with pretty strict punishments. That was pretty firmly established by both DS9 (in "Doctor Bashir, I Presume" among other episodes) and Voyager (in "Lineage"). And we saw (in TNG in "Unnatural Selection") what happened when the one human genetic engineering lab we've seen in 24th century Star Trek tried to engineer away disease through the development of a proactive immune system; just imagine how bad things could go if they tried to engineer away mental and social problems.
Also, you're putting a lot more on genetics than there actually is. It's almost certainly not going to be possible to "breed out" or engineer away psychological maladies or social dysfunctions, because those, even while likely having a genetic component in increasing the tendency towards it, aren't wholly genetic, but are in large part the result of learned behavior and experiences. Not everything about people is in the DNA, and there's some things that even infinite knowledge about DNA couldn't prevent.
And there's even an argument to make that trying to remove such things would be decreasing the psychological diversity of humanity to a potentially harmful level. Setting aside social problems, since those are absolutely completely independent from and unrelated to DNA, and just focusing on the psychological: Mental disorders for the most part aren't like biological diseases in the sense that they're a foreign or abnormal thing imposing themselves on a person. There are a few examples of mental disorders like that, yes, usually the result of brain damage. But most of them are instead just extreme examples of traits that in weaker levels are normal and even beneficial. They're more akin to, say, a fever; something that at a healthy level is a normal reaction to stimuli, but that in certain situations goes too far to the point of risking the well-being of the person experiencing it. They're the brain taking a pattern of behavior to an extreme for one reason or another, and you can't completely remove that possibility without snipping out that pattern of behavior entirely. Which is exactly why the therapy side of treatment nowadays isn't about curing, but about managing; about learning to adapt to patterns of thought so that you can deal with them in a healthy manner, rather than trying futility to stop the patterns of thought entirely.
You likely can't fully remove paranoia from the human psyche without removing pattern recognition, you likely can't remove anxiety without removing adrenaline responses to stress, and you likely can't remove psychosis without removing the ability to form counterfactuals. Even removing genetic tendencies could have unforeseen side effects in limiting the healthy mental development towards certain ends. What you can do is treat such things when they appear through therapy and medication, to allow people that suffer such problems the ability to function and thrive in society.
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