That whole supposition seems to me as groundless and unsupported as the one that tries to dispute. Even more unsubstantiated because it is contradictory to empirical evidence. (And no, as far as I’m concerned, Kelvin film & DSC ships can’t be used as corroborations and don’t help us reach for safe conclusions about TOS era Starfleet.)
Forget what @Crazy Eddie says. He's just plain wrong. There IS evidence. The TOS-movies mainly. There are a vast number of ships visible, with different mission spectrums, as well as further hints like how many ships are available at any time/how big the doors of a space station are/what type of starships we see protecting Earth.. etc. etc.. Also everytime other starships were seen/mentioned in TOS, including fleets. The "big gun" was always the Constitution class, until the Excelsior came around. And during TNG the Galaxy was obviously the newest and biggest one. Crazy Eddie simply mismatches personal headcanon (his own) with official canon. Not the first one to do that. But a little obnoxious since he can't differentiate between the two.