I'm not trying to make light of your experience, and I do appreciate that you care about people you have to look after deeply.
However, I also don't think what you described is professional with respect to being a manager. A manager's job is to balance the interest between stakeholders and their reports, make sure things happen as promised (and adjust scope when promise can't be met), and make sure everyone in their team grows (skill, remuneration, etc.).
> Cancer/long-term illness, death in the family, miscarriages, etc all weigh heavily on my mind when something happens to one of my direct reports.
These are not things that would affect you from being a good manager. That's just part of the job of being a manager. It's a different kind of problem, it's not necessarily heavier problems. It's unhelpful, or even narcissistic, to think that you're actually quietly taking on more burden for everyone without being appreciated.
You should absolutely have empathy as a manager, but if your empathy is weighing on you instead of enabling you to be a better manager, then you're probably not doing a very good job as a manager.
> Not asking for pity, but being a manager has a heavy emotional toll if you have even the slightest bit of empathy that I don't think most ICs appreciate fully.
I disagree. That's just an excuse of not doing what you should be doing as a manager: manage things. See above.
If you are an empathetic person, if you work in a team where people have/had "cancer/long-term illness, death in the family, miscarriages, etc.", then it's emotionally just as taxing whether you're an IC or a manager. In fact, if you can't manage your emotions and get what needs to be done done (damage control, repriortization, etc. to get things shipped), then you're not doing your job as a manager.
Edit: feel free to downvote if you don't agree but much appreciated if you also leave a response to say why you disagree. Otherwise, I hope you feel good "punishing" people just because you feel differently but don't actually have a concrete argument against.