Humans within the Federation probably do believe the whole evolved sensibilities thing and aside from a few border skirmishes with the Cardassian and Tzenkethi TNG was during a time of complacent peace and Picard I don't think fought in the border wars so he had this boy scout mentality."I'm a soldier, not a diplomat." -- James T. Kirk.
It's funny. Even today, some folks still talk as though "Star Trek" = TNG, but honestly, at this point TNG is a more of an outlier when it comes to aggressively pushing the whole "evolved" human utopia business to the extreme. And, in its defense, even TNG eventually allowed for a more complicated, realistic portrait of human nature in episodes like "The Drumhead" and "The Wounded."
The way I see it, you can't really explore the human condition by declaring that humanity has "evolved" beyond all the uglier, messier parts of humanity.
Heck, the whole point of "The Enemy Within" was that Kirk needed both his positive and negative qualities to be whole. He couldn't just discard his "evil" half.
Humans will never have their shit together, as Quark pointed out take away their comforts and they are as vicious as the most bloodthirsty Klingon.
As you pointed out episodes like the Drumhead (a personal favourite) chipped away at the cornier declarations of utopia. So TNG starts to come across less like humanity is evolved and flawless and more humanity is trying really hard to champion its ideal image of itself but then reality hits back and things like schools and children on starships stop being a thing. Barring the occasional child born in space with nowhere else to go of course.