EDIT: Did a little research.

You are correct that a fire appears to have affected many of the records of those discharged up until the year 1964, which would affect all Korea vets and a significant portion of Vietnam vets. However, the records people themselves said they can still confirm whether the vets actually served, even if they can't recreate specific details of each individual record.

https://www.archives.gov/news/articles/archives-recalls-fire

Bryan McGraw, access coordinator at the NPRC, emphasized the gravity of the loss of the actual primary source records. [...] “We can usually prove eligibility for benefits and get the vet or next of kin their entitlements; however, we cannot recreate the individual file to what it was—we don't know what was specifically in each file, and each of these was as different as each of us as individuals. So from a purely historic or genealogical perspective, that material was lost forever.”

They were apparently able to cross-reference a lot of other records to prove that people served.

It's just my experience that when an alleged vet is both making vague claims as well as claiming their records were lost or highly secret or whatever, it's a bullshit story. The military will confirm whether or not someone served, and usually in what capacity, and usually over what time frame they served. I don't think the fire erased these people from the military institutional knowledge and makes them impossible to find via a regular FOIA request.