First of all, what is meant by "family show"? Ask 10 different people and you will get 10 different answers.
I think the relevant question is how the term was understood by network programmers and censors, since they were the ones who had actual influence over the content and scheduling. The term was
not strictly a matter of opinion, but a matter of policy. As I mentioned before, there was a brief period in the mid-'70s when Congress actually required that early prime time be set aside for family-friendly programming, and though that was soon revoked after a free-speech lawsuit, it remained standard network practice through at least the '80s to schedule family-friendly fare at 8 PM and save the more adult stuff for 10 PM when the younger kids had gone to bed.
On a practical level, what did this entail? Basically don't show overly graphic depictions of sex and violence or overdo the language. Let's say that even the limits imposed by the FCC (at least in the 80s) did not allow much room for maneuver.
That's part of it, yes. Also, family shows tended to tell lightweight, purely entertaining adventure stories without a lot of depth (e.g.
Knight Rider), or they told wholesome, sentimental stories with positive messages (e.g.
Starman).
Bu there were some problems. Science fiction in general and Star Trek in particular was considered "kid stuff." Even the rating at the time (since changed) was a fairly harmless "TV-PG".
That system wasn't instituted until January 1, 1997, so there was no such rating for ST:TNG "at the time," only retroactively.
Here comes Berman who takes care of "smoothing out" some of the excesses of the Roddenberry period. So the characters talk less about sex, the violence is less explicit.
However, when it comes to age, the target remains the more adult one. No one ever said that the seasons of Star Trek after the first two were more "kid-ish." If it is necessary to make it clear that a character has had sex, it is made clear, even if in a less explicit way. People keep dying. Topics usually considered "sensitive" such as religion are addressed.
So, more than a family show, a show where parents don't have to worry too much about whether or not to watch it with their children. If the topic covered is complicated for a child to understand there is no problem, as long as there is a spaceship that goes "Fwwwwuuuuusshhhhhhhhhhh".
That's fair, though I'm not sure kid-friendliness was as much of a consideration as you suggest. I think Berman and Piller just weren't as invested in pushing the sexy stuff as much as Roddenberry wanted to. I could be stretching here, but I think the shows and movies at the time that did play up sexual content were often seen as trashy, and the "Jiggle Era" of 1970s TV (exemplified by shows like
Charlie's Angels and
Three's Company) was still in relatively recent memory. So maybe taking a more sedate approach to sexuality was more about being respectable and classy than about being kid-friendly.
This brings us to LGBTQIA+ representation, or rather the lack of it. If in the late 80s/early 90s it might have made sense from a cynical business point of view, after that much less so, especially for a show like Star Trek which prided itself on being at the cutting edge in dealing with even uncomfortable topics. Then in 2005, when they aired The L Word and on the same network as Enterprise there was Buffy, where there was a lesbian couple, it was completely incomprehensible.
Why was this happening? We don't have a precise answer and probably never will. Berman simply didn't want it, even though fans were asking for it, actors were asking for it, and writers were pitching stories with LBGT themes. When asked specifically why, he responded with a confusing word salad.
Did I summarize correctly?
Pretty much, yeah. But you left out the part where Roddenberry overtly promised fans before TNG premiered that it would have gay representation, and then the show reneged on that promise. So the argument that it's an issue that only came up later is simply wrong. It was on the fans' minds from the beginning, and it was explicitly promised from the beginning. Part of the reason TNG's uncredited co-creator David Gerrold left the show so early was because his attempt to write an AIDS-allegory episode with a gay character was shot down by Roddenberry's homophobic lawyer and others. So this isn't just an issue that came up later in the '90s or '00s. It's a pledge that was made and broken right at the start.