I've long felt that Insurrection would've worked better more tied to the Dominion War or putting a Dominion War TNG film in its place. As for the pushback that the average audience wouldn't understand what was going on if plopped in the middle of a Dominion War film, I think there have been many war films about battles and not the war itself. Either in dialogue or written at the beginning of the film could set it up what was happening. If this focus is just on one battle, one planet (thinking like the novel The Battle of Betazed) then I think the audience for the most part can grasp what's going on, if nothing else the personal stakes of the characters, if not the wider galactic stakes. Further that might funnel new fans to look at DS9 to see what happens with the Dominion War after the film. (I recall the X-Files movie, Fight the Future. It did well, from what I recall, without anticipating that everyone who looked at it was steeped in that series' alien mythology). As others alluded to, how big was the 90s filmgoing audience for Trek anyway? First Contact did gangbusters at that time, but that would be relatively peanuts today, so the audience was somewhat limited already. Further I think the goodwill First Contact had engendered perhaps could've gotten whatever new or casual fans that liked First Contact to give a more inside baseball Trek movie a chance or be patient with it.
A few other things about the Dominion War TNG movie...for any Trek film there are going to be bad guys, so why not use established bad guys? And using the Dominion while DS9 was on builds up the whole franchise, makes the experience all the richer. When I look over other Trek movies, didn't seem people were too put off by the return of the Klingons (without real explanation for who they were and what they meant on TOS) and most definitely Khan (who had been in one episode almost 20 years before TWOK). The writing, acting, special effects, and direction got audiences into TWOK without a primer. I believe a Dominion War TNG movie (looking at a story like Battle of Betazed or the TNG Dominion War novels) could've done it. Battle of Betazed makes the stakes personal for Troi and also allows for a return of Troi's mother, and the TNG Dominion War novels is just one big mission inside of the war, but one that would have stakes for that particular film but also DS9. It would've been great as well if they had also been able to have Ro in a film.
For the original poster, the idea of Shinzon being a Changeling is inspired. However, I do think that undercuts the philosophizing about nature versus nurture because a Changeling Picard is just another being impersonating Picard versus a younger clone of him who is struggling to establish his own identity. I also don't see the need to have the Remans be another genetically engineered species, unless they are changed to an improved version of Jem' Hadar. But still the changes made would make putting Earth in peril make a lot more sense than it did in the actual movie.
For a good while, on various threads about Nemesis, I've posted that it would've been better to have Romulus in peril and that the Enterprise-E risks all to save the Romulan homeworld. I think the creators sought to emulate the wrong TOS film for Nemesis. They were looking for another Wrath of Khan when The Undiscovered Country would've been a better way to go. And I would've preferred they brought back Sela and Tomalak in some way, even if in cameos.