It's been awhile since I read the final empire, but didn't he start rapidly aging as soon as he took the bracers off? I think like with weight, your body has a set value it'll go to when you're not tapping. So when he's not tapping his atium-age reserves, he ages. And since he's been around for a thousand years, he has to become old while storing up new age just because of the amount he has to tap to stay young.
However, since a body naturally heals itself and tries to get from sick to healthy, healthy seems to be the default state when not tapping (and when not affected by a disease or injury). It doesn't take as much tapped gold-health-energy-stuff to stay perfectly healthy, since his body is already healthy (from previously tapped gold-health). Because of this, he can use his massive amounts of stored backups to fill new gold and burn it (to restore reserves).
To sumemrize: the body "knows" it's age and will return to it when not tapping age. The body, however, doesn't "know" it should be unhealthy. Thus, it doesn't automatically grow sick when not tapping. It'll always take the same amount of tapped gold-health to stay healthy, whereas the amount of atium-age required to tap to stay young is CONSTANTLY increasing.