As an ex-AF uhm, person, (Not gonna use the word Vet since hell, only 23 and makes me sound old as hell), I agree. I had asthma as a kid, I no longer have it. Didn't tell them. Never became a problem. I told my recruiter I had tubes in my ears when I was a baby and hardly remember it and he told me "You don't remember" so I repeated what I said thinkin it was some joke and he said "No, you don't remember". Had physicals, got sick and saw the doc, they asked me "Ever had tubes in your ears?" "Yeah, when I was little, I don't remember though" and that's the end of that story. Nothing ever came of it. Military truly doesn't care as long as it doesn't affect you. If you can run longer than what the PT test is without any problems, you're fine.
They don't go out and dig up every record about you. They'll look to see if you have any criminal history and stuff like that but as far as "Oh, John had asthma when he was 6. He hasn't been treated or anything for it since he was 8. We'll flag this"? Yeah, that definitely doesn't happen.