I don't think this crosses all designators at all.
Medical care is not uniform, and I would go so far as to say Shipboard Sea Duty probably receives the lowest quality of medical care.
If you're ship's company, and even worse, not on a big deck, an E-6/7 IDC is the person who makes the determination whether or not you need higher level care. This isn't to say they uniformly suck, but if you really need care, you're going to have to be pretty persistent about it, which may be hard for an O-1/2 to figure out, especially without anybody to help them out.
Having said that, I have seen shipboard CoCs pull people guys from Physical Therapy regimens to get underway because they held quals that were redlines, and they didn't want to deal with the hassle of trying to pull guys from other ships to meet tasking.
And from the SWO perspective, there is absolutely going to be a tendency to put quals first if the issue isn't flat out disqualifying you from sea duty. Not qualifying on time can seriously screw with your career.
On shore duty, I've had little trouble getting an M.D. if I wanted a second opinion after the Corpsman's diagnosis.