Two thirds into TNG I'm now really frustrated with the story being told. Calling it 'a bit conservative in places' is nothing new or controversial, but to me it's as if the framing is all wrong: the show is extremely and explicitly conservative by default, so it doesn't surprise me at all that conservatives like it. Why then do "progressives" gate-keep it as a thing that conservatives won't/shouldn't like?
Assuming Ian Danskin was on the money in his essay on the origins of conservatism, then TNG can be framed as the answer to the question, "what if we did conservatism through a mechanism other than the market?". Without the market, the show can throw out surface-level progressive ideas like not using money, but conservatism was never inherently capitalist so being anticapitalist isn't inherently progressive either. Likewise, stratifying people by race and gender is gone, but the stratification is still there, now achieved with weird maths and science exams. We're not racist now, sure, but we're hardly done discriminating.
The actual ideals of conservatism are things like a strict social hierarchy, and a system for dealing out power to the deserving. Both of these are handled by Starfleet. Starfleet separates society into 'officers', 'crewmen', and 'everyone else', with officers clearly at the top. The chain of command is omnipresent, and progression up the ranks is at the discretion of those with power. Starfleet maintains incredible influence over the Federation, disallowing any other militaries within its borders and making most of its decisions (up to and including starting wars) without consulting any government. The proving ground is the bridge of the starship on the final frontier – do well and you'll have a ship of your own real quick.
The point of all of this is to give the powerful a means to maintain their power, so of course nepotism plays a role; poor Wesley Crusher, as the son of two officers, deserves the early access to the bridge as an acting ensign (and of course he'll grow up to be Mozart so it's justified). Ro Laren is later afforded the same privilege to the initial enmity of Picard and Riker – she is, after all, the daughter of dead nobodies. Social mobility isn't dead though: if you're as clever as Picard was as a child you too can captain the flagship (and if not, you can always stay a grumpy twat in the vineyard before dying in a fire offscreen).
This sucks. TNG is trying to persuade me that conservatism makes sense by giving us Picard, the paragon of virtue, and a universe in which he and his crew of geniuses can always make the big decisions and save the day, because the undemocratic aristocracy that put them there works actually. Unless you're Picard, there's nothing progressive about it.