The way I've seen it actually work (and I have been both PM and PO at the same time) is that the PO gets epics and possibly stories from the PM or director, and is responsible for fitting the most important epics into the next sprints.

Yes, it includes bs like being able to on the fly answer or estimate, when will this ticket from two months ago. And if you don't give an answer right away you're "not as on top of things as I would like".

PO is a garbage role. And in practice it is focused on backlogs and standups and sprint planning.

Usually PO is there for the non technical PM to not have to talk to engineering.