That’s my recollection of it. I think there was very little, if any, draft between about '54 and '65. That point isn’t terribly important to the OP’s scenario, but it would be unlikely, although not impossible, that the character would have not been in during the Korean war, if he were born in '28, or that he would have been allowed to enlist, for the first time, in the mid '60’s when he would be in his mid thirty’s.
The assumption is if you can’t get promoted you are a dud. That is not always a valid assumption. Someone who is a great squad leader may not make a great platoon sergeant or first sergeant. There are different skill sets for each position. It has always been the military way to promote to the level of your own incompetence. If you want a good account of this procedure read David Hackworth’s autobiography About Face.
The other more valid point is that if you keep in more crusty old NCOs it adversly affects the young motivated troops that are below them. There won’t be slots for them to get promoted. That is especially true in the Corps which is much smaller than the Army. See what the military was like before WWII when you could easily retire after 20 years as a corporal.
The 'up or out" regs. came in the 70’s. Prior to that it wasn’t particularly unusual to see an E-4 or E-5 w/ 16, 18, 20 years service. Even in the officer corps. things were a lot more lax.
Indeed so, that would be a very good fit – and he does a relatively long tour as GYSGT and “retires” to reserves rather than obligate himself to more service by going for the by-now due promotion.
In the case of the officer corps, my sources indicate that even in 1960 they had an unofficial up or out expectation. The way it was explained was that under certain circumstances an officer is considered to be “in the zone” for potential advancement. If this happens three times without a promotion, then the officer was expected to resign.