I absolutely appreciate everyone's input and I think I've settled on my preferred ruling.
In a home game, I would have no problem ruling that skeleton, ghost, etc. PCs don't bleed. They're a Rare ancestry so I could disallow them altogether if it was really going to be a problem.
But I think RAI and RAW that PC rules are different from 'monster' rules. All PCs breathe, bleed, eat, etc. UNLESS stated otherwise.
I think mostly that comes from the "Too Good to be True" rule.
If skeletons don't have blood, they can't bleed. Since they don't have lungs, they don't breathe. If they don't have ears, they are immune to auditory effects. No organs? Then they're immune to sickness.
Once you start picking apart aspects of the player character, you can start justifying other things. I know that might seem a bit like a slippery slope argument but some similar points have already been brought up in this thread - I don't think it's that unreasonable to expect other rule changes to be requested once you start making exceptions.