On the other hand, I think they both see their bodies as their most effective weapon and would want to keep them functioning at peak performance; seeing a doctor would just fall under weapons maintenance.

Yes, this!! Honestly I'm clueless on how they would approach medical professionals but I feel like they would be kinda like the rest of us (or some of us) who just don't like hospitals and MDs at all and won't go unless it seems like something that won't take care of itself. As for ducking out early, I'm actually wondering if SHIELD itself would sometimes try to get them to leave early in order to complete a mission and the MD's not happy with it.