Look at Discovery--in that case I would argue the changes made were much more significant, changing the entire focus of the show.
Of course, but those shifts in story emphasis didn't create continuity questions like "What the hell happened to the civilians and families that the ship was literally designed to accommodate?" That's the root of my problem -- the whole design of the
Enterprise-D and the fact that it had civilians on board in the first place was the result of it being intended as a long-term deep-space exploration vessel that would be away from the Federation for up to 15 years at a time and thus had to be a self-sufficient community in its own right rather than just a military vessel, and therefore needed a way to protect that civilian community and take them out of danger when the need arose. There are a lot of features about the ship itself, its design and crew composition and such, that just don't serve a purpose anymore if you take that premise away. It's not just
that they made changes -- it's that the specific changes they made created logic holes.
I guess I do wonder what the show would be like if saucer separations and the civilian population hadn't been sidelined (let's assume that the former were done in a way that didn't bog down episodes/kill the VFX budget).
Budget wasn't the issue. Once they shot the separation sequence, they could've reused it as stock anytime. The problem was that the 6-foot miniature was too big, heavy, and awkward to work with. If they'd just given the 4-footer separation ability, or built separate saucer and stardrive models along with it (an added expense but only a one-time one amortized over the rest of the series), they could've shot more sequences of the separated ship about as easily as they shot new footage of the intact 4-footer.
Would we remember it as (generally) fondly as we do now, or would the show have been cancelled after bleeding viewers because continuing down the path laid out by the first two seasons proved untenable?
I don't think the one has anything to do with the other. The show got better because the character writing improved. That could certainly have still happened if the focus on the hybrid Starfleet/civilian makeup of the crew had been kept; indeed, it could've created even more opportunities for character drama and conflict because of the different points of view of the two groups, and the debates that could've been had over the ethics of bringing civilians along as the galaxy got more dangerous. We saw how good the writing was on DS9 with a mixed cast of Starfleet and civilians -- Michael Piller could've done the same years earlier on TNG if he'd wanted to.
I've often felt they should've played up the civilian element even more. Maybe have Deanna Troi be a civilian, the official liaison and spokesperson for the civilian crew, hence her participation in command decisions and her lack of a uniform. If one of the main cast had been a civilian, it would've given the writers more of an entry point to write about them.
Then there's my idea for a more radical reinvention, which I may have already mentioned. Instead of having a separable saucer and battle section, have a civilian research ship commanded by Picard (in the vein of his real-life inspiration Jacques Cousteau) escorted and protected by a Starfleet ship commanded by Captain Riker. There could've been some interesting conflicts between the two captains and crews and their differing viewpoints.