The term "flagship" strictly speaking, should mean either the ship from which an admiral commands or the lead ship in a task force or armada. But because most ST writers are laypeople rather than naval veterans, they tend to use "flagship" more in its vernacular sense of the most prominent or special member of a group, like how Voyager was the flagship show of the UPN network, say. So the idea is that the Enterprise is the jewel of the fleet, the most advanced and prestigious ship with the best and brightest crew.
Which I happen to think is a lousy idea, a sort of elitism that doesn't seem worthy of the Federation. Also, doesn't it make more sense to distribute your best personnel throughout the fleet instead of letting one ship hog them all?
Not to mention that if the Enterprise-D went down in battle, then the fleet's so-called "best and brightest" would all go down with it.
But the problem is that the attitudes of fandom have been transposed onto the characters in-universe. We care about the Enterprise the most because it's the ship the series focuses on, so that's led to it being portrayed as the most important ship in-universe too. Gene Roddenberry probably started this with his TMP novelization, in which he claimed the E was special because it was the only ship to return home from a 5-year mission with both ship and crew essentially intact.
Oh definitely. One of my favourite things about TOS was how Enterprise was 'just another ship', it wasn't seen (in universe) as being anything more special than any other. But as you say, Roddenberry's 'fetishization' of the Enterprise in-verse in his novel of TMP seemed to open the floodgates towards her being seen as a 'special' ship, and her crew as 'living legends'. IMO it actually makes them less interesting.