Sounds like a great idea.
I worked 5 1/2 years in a NYC hospital. Used every sick day. We considered it mental health days. And as you know in nursing sick calls leave everyone there very overworked. It's like a vicious cycle.
I came to a suburban not-for-profit hospital and tried that. I was put on probation. Especially calling in on a Monday, Friday or weekend. A tough supervisor set me straight.
Before I left 15 years later I had 5 months of sick time (considered it like insurance) which would have been nice as some kind of compensation at the end. At least I left with the knowledge I didn't screw the place and have a good reputation.
I certainly can relate. Our hospital (large tertiary care center in a rural setting) had almost the same atmosphere as your NYC hospital; mental health days were taken liberally. When they implemented the new PTO/ETO system, it changed the nurses' way of thinking about sick time (as well as all of the other employees). I'm thankful now that I have the sick time built up, because now is when I really need it.
I think there's a lot to be said about your last statement; your conscience is clear because you didn't abuse your sick time, plus you have a better reputation for having not abused it (I recently had to take 3 days off related to my chronic illness; everyone was talking about it, saying "boy, he must REALLY be sick if he took sick time").