The man who is sick should do what he is able to do. If he really can’t daven, then he is “annus” – someone who was unable to do the Mitzvah. But if he can at least daven the minimum that is required, he should do so. I would assume that in that situation, most men wouldn’t go to shul, but would daven 3 times a day. And they probably only do the basics and skip the parts that are not absolutely required. (how much they would skip would depend on how sick they are and what they feel they can do).
Before someone takes a job or enters a field that does not allow for davening, he should ask a sheilah as to whether or not he is permitted to take that job.
If being an athlete, pilot, or truck driver means not davening, I would guess that most Rabbanim would tell most people to find a different job.
For a doctor, I would imagine there would be leniencies since he is involved in pikuach nefesh (saving lives).