I'll say this for this thread - in spite of its often frustrating circularity, it's given me a lot of food for thought, particularly in regard to how I would define "family friendly", and whether I'd put Trek in that category.
Upshot is, now, I wouldn't class any Trek series as strictly "family friendly" other than TAS and Prodigy, and thus am recalibrating my own relationship to the show a bit. For one thing, it gives me no excuse to ever refer to the current streaming era as "not Star Trek", because it is Star Trek, just a Star Trek that doesn't appeal to me personally, and that's an end to it.
For a UK perspective, TNG was broadcast at 6pm on BBC2, Monday to Friday, mostly repeats, mixed with repeats of TOS, and once a year a run of new stuff. The only variation I can recall was, as Relayer1 noted, the frustrating frequency with which it was co-opted for sport. I don't think there's much doubt the BBC saw Trek as little more than lightweight filler, something bright and harmless to fill an early evening slot. That's very likely why Conspiracy prompted controversy and censorship, and I wouldn't be surprised if other eps did too, like Genesis, as what (presumably) proto-Worf did to the Sacrificial Ensign Of The Week was fairly gnarly.
The UK tends to be stricter on violence and blood than the US, and more relaxed on nudity and sexuality, thus our definition of "family friendly", and by extension mine, is a little different. For instance, nudity can and has appeared on UK TV before the 9PM watershed, and in the daytime, it just has to be educational, or artistic, or otherwise entirely non-sexual, and brief. One example would be The Simpsons Movie, broadcast multiple times in the early evening - around the same time as TNG was, funnily enough - on Channel 4 with nothing edited out. That's not to say the BBFC and the broadcasters can't be as opaque and arbitrary and contradictory as the MPA, because believe me they can.
As to the matter of LGBT+ content, in response to the seeming suggestions that a "softly-softly" approach is the best one, I'd counter that those now spearheading aggressive rollbacks of LGBT+ rights and representation in the US, UK and elsewhere were very likely emboldened by that very approach, by having little to no firm pushback. Strong and clear is the only way to be, in my opinion, or this is where we end up.
Further to that, and noting that kids shows like Grange Hill and Byker Grove were tackling LGBT+ characters, and quite well from what I remember, in the UK at around the time TNG et al were mostly pretending we don't exist, I'm firmly in agreement with Christopher that Trek, and Berman in particular, seriously dropped the ball, and I struggle to see that as anything other than deliberate on the latter's part.
One last thing that caught my eye. Someone. I think Oddish, brought up two recent Disney films that failed, and strongly intimated it was because of their LGBT+ content. I'm guessing those films were Lightyear and Strange World. If you honestly believe the former failed thanks to a blink-and-you'll-miss-it dab of the lips between two women, and not because it was an ill-conceived, sloppily executed muddle, I think that says a lot more about you than the film. Equally, Strange World, at least in my opinion, suffered far more because of poor publicity - flat, generic title, dull trailers - than happening to have gay romance in it. The queer folk weren't the issue, it's Disney's recent inability to properly sell their films, and ever more executive-controlled approach to making the films in the first place. Wish is the perfect example of this; a Disney film by numbers poorly advertised, and that landed with a thump as a result.
Feels like there's some pretty disingenous stuff going on in this thread, and it's helping no-one.