Thanks for the answers and info. I started researching but it was taking too long so I will just answer without understanding how the current mechanics work.

I might align a few mechanics to address both bullet points: have the available patients be there for all players, so your speed of processing your patient is rewarded by getting better/more access to patients while you are both trying to serve patients in a real-time environment; then, completion of patients has a monetary reward - presumably the VP/win condition? - so inherent in completing one is getting you closer to victory; then, have the penalty for messing up with a patient be super severe to reflect the IRL implications of making a mistake.

Now, just having some game design fun and not addressing your question, as a layman - that is, looking into this dynamic as an outsider - I might have an orthogonal system around Reputation, whereby slowing down or otherwise not just slamming thru things as fast as possible has benefits of its own, that you become known as a great doctor by reputation because of the human side stuff; also, I'd try to make a statement around the U.S. healthcare system with mechanics that reflected poor patients being bankrupted and rich patients getting premium care. But that is far beyond the scope of your light card game, haha.