The next apply only to evidence we are aware of, even though it is always possible that there would have been events that were never televised, filmed, or even alluded to in canon.
1) Did anybody call the ship a flagship of any sort?
2) Did anybody call the ship "the" flagship as opposed to "a" flagship?
3) Did we learn what the ship was flagship to?
4) Was the ship a flagship at any point, in the sense of flying the flag of a flag officer?
5) Was the ship a flagship at any point, in the sense of commanding a large task force?
6) Was the ship a flagship at any point, in the sense of being a symbolic showpiece vessel?
Kirk's TOS ship: no, no, no, yes, no, unknown/unlikely
Kirk's refitted ship: no, no, no, yes, no, unknown/unlikely
The E-A: no, no, no, no, no, unknown but possible considering TUC
The E-B: no, no, no, no, no, unknown but very likely considering GEN
The E-C: no, no, no, no, no, unknown but somewhat possible considering relative size
The E-D: yes, yes, yes (of the Federation), yes, yes, yes
The E-E: no, no, no, no, yes, unknown but possible
nuKirk's nuship: yes, no, no (beyond Pike's "our"), no, perhaps (at departure to Vulcan, although there's no dialogue to that end), unknown but likely considering modernity etc.
So, out of the lot only the nuship might be "The flagship of Starfleet", unless one of the others briefly acquired that status off screen and we were never told.