What's worst was the regular appointments, "Bob on Thursday morning" or "Martha's weekly accupuncture treatment for MS." You get to know the people. You (I) can't avoid developing bonds and caring. You either watch them go on with the treatment not making any improvement or, the worst thing, showing up one morning and being told "oh, we're sorry, we forgot to call you and cancel..."
I understand exactly what you're saying. It's so much worse when you get to know them. And their families... that's even worse. Burnout is big in those professions and you gave the EXACT reason why. :-(
The one thing that I got from my time in the trenches that is still with me is my wife. In addition to my work on professional ambulances I also did time on the volunteer ambluance in my home town. For six months I was the crew chief on the Friday night shift (no one else would take it). And the woman who is now my wife and her mother were my crew. Of course I got spoiled as my wife did everything I told her... then we got married... ;^>