Interesting axiomatic basis for your argument, and thank you for explaining where we are seeing differences. You are correct that I would disagree with you on that, and place both as necessities. Sick days are necessary for short-term physical and mental health. Vacation days are necessary for long-term mental health.

If you missed weeks of work from an illness, those weeks aren't time where you could see family, see the world, relax, and be human. They don't satisfy the same long-term needs as fully-disconnected vacation time satisfies.