Technically, that is true. The Enterprise became a flagship when Kirk took over command in TMP. It ceased being a flagship when he was demoted to Captain.
Umm, except that Kirk made a big deal out of not really being an Admiral while in command of the ship in TMP. He even changed his rank braid down to Captain.
We know Starfleet believes in the concept of flagships as command vessels: the terminology is used this way e.g. in "Descent". It also appears Starfleet or the Federation uses the word in another meaning, by calling the E-D and later the E-E the Federation Flagship - apparently denoting the leading, shiniest, most representative vessel in Federation service. Neither of those meanings of the word would seem to apply to Kirk's first Enterprise, however.
Timo Saloniemi