Not always... pretty sure even for the brief time I was in a public primary school here in Australia my parents had to contribute for textbooks. I suspect they had some kind of aid available for families who really couldn't afford it, but I'm not sure.
But the entity who decides to use the book is pretty much the same as the entity who budgets the money for the book. In a university, the students pay while the professors/departments choose.
I see your point, but I think individual schools don't make the choice. This stuff is usually done at the school board level, or maybe even State level.
My sister's public middle school in Indiana has student book fees, and various fees for using mandatory assigned Chromebooks, some of which are refunded if it's returned in good order at the end of the year, some aren't.
If you are low-income, you can get the fees waived, and the majority of her school books are still free. But not all.