How would any 1701 crew serving on Discovery never mention that experience in TOS or the movies?
It never came up? We know Kirk was on the Republic and then the Farragut (until 2257), but we have no information on what assignments he had between 2257 and about 2265. Discovery could've been one of them.
Spock's background is pretty well settled, but we know nothing about McCoy's background in Starfleet. Scotty served on 11 starships in his career: we know of three. Not to mention the hundreds of ancillary characters in Star Trek who we've not been privy to their assignments circa the 2250s.
With one-time random Klingon Houses from canon being namedropped and expanded (House of Mokai), I expect the same to happen to one-time random mentioned characters from TOS.