#1 I'm not saying there's evidence the Ent-E "is" the flagship, I'm merely speculating it would make sense, given it's the same crew with a new ship, smaller, but just as capable.There's also no positive evidence that it was, as it was never referred to as such and was specifically per dialogue excluded from the post-Dominion War negotiations (and indeed evidence that it was even part of the War is semi-canonical at best).
OTOH, several Galaxy-class starships, including the Venture did serve during the War and in a flagship role in the ordinary sense.
Why take it away from it just to reassign it to Enterprise. I mean, we see the Enterprise lineage as special because those are the stories that we've seen, but other ships/lineages could have easily been the "Hero of Another Story".
Why not? There have almost certainly been Federation flagships not called Enterprise in the past. FWIW some official social media suggests that a late 23rd/early 24th Century predecessor (NCC-1777, Shangri-La-class) served as Federation flagship while the Enterprise-B (and Excelsior who I've always thought was a reasonable candidate) was in service.
More "bad fan service" rather than "bad fanfiction" IMO, but I agree
Admiral Beverly C. Crusher (nee Howard), MD. At last report was serving as Head of Starfleet Medical* and heading up the De-Borgification/Counter-Changeling Task Force.
* Either the Head of the Medical Branch as a whole (A post typically held by a senior flag officer IRL as they manage 100-1000s of physicans, and known as the Surgeon-General, Starfleet certainly doesn't have less and Starfleet did apparently have an SG during TOS) or merely the Head of the location known by the shorthand of "Starfleet Medical" (in full the Starfleet Medical Annex - San Francisco), which would be much more inline with her displayed rank of Commodore (and indeed her having held that post as a Commander or Acting Captain in the past).
#2 It's been a while since I watched DS9, Venture was that one Galaxy-class ship we kept seeing docked at DS9 and in fleet battles? I kind of like the idea that Starfleet assigned a Galaxy-class ship to lead the war front. I also liked that we saw several Galaxies fighting the Cardies and Dominion in the final seasons.
#3 If Starfleet wants Picard and crew to maintain flagship status and carry that over to the Enterprise-E, it's possible another ship could have carried the mantle for a year or two between Enterprises, then moved on to it's next mission, whatever that would be.
#4 I pay zero attention to Star Trek in social media, unless you count TrekBBS and YouTube. I just go by what's on screen in the TV shows, the films, and the behind-the-scenes stuff on the DVD's, Blu-ray, or stuff that pops up on YouTube and in online articles or Memory-Alpha, you get the idea.
#5 Bad fan-service, I guess, but yeah, I'm glad we agree on this. haha
#6 Is this a fan-fiction thing? There was Captain Crusher in the anti-time future of "All Good Things..." and retired Dr. Crusher in PIC S3, but "Admiral" Crusher?