It's well played, even if probably unintended. I was quite surprised to see the ass kicking going on.
I sure hope not. That would be rather impractical, especially since the Enterprise often faces major threats and destruction.
VOY and the Defiant encountered similarly major threats and were faced with destruction (the Defiant was in fact destroyed once before during the Dominion War).
Point being, we don't have evidence to confirm or deny the premise that the 'honor' of being a 'flagship' won't be temporarily (or permanently) assigned to another vessel.
But if I may ask, what would be the difference even if the ENT is not the flagship?
Quite literally nothing changes. Sure it may deal a blow to morale, but come on, even those people would have to understand that various circumstances can destroy the flagship of the Federation just as easily as another starship can be destroyed.
Remember how the Ent-D went out? If anything, Riker should have been court martialed for not 'firing everything' they had at the BoP (which would realistically end up obliterated in a full front assault from a galaxy class starship, shields, or no shields).
ENT-D destruction would be more in line with 'target practice' analogy rather than a Living Construct that infects an armada of SF ships who then start attacking each other with the purpose of destroying themselves entirely and infecting new ships as they Warp in.