This is not generally true in the USA, where the "first sale doctrine" gives libraries (or anyone else) the legal right to buy an ordinary consumer copy of a physical printed book, and lend it out (for free or rent) as many times as they want, without the permission of the copyright holder and without paying them an additional fee.

In the USA, if any library pays more for a "library" price, that's voluntary. They have the complete legal right to buy an ordinary copy from any consumer channel that will sell ot them (new or used re-sold) and lend it out. And this is not a special right of things classed as libraries, it's a right anyone has. Traditionally, when it comes to physical copies, anyway.

In other countries, the "first-sale doctrine" does not exist, and this may not be true. I think in Europe libraries do pay "licensing" fees to be able to loan out books, that get redistributed to copyright holders based on use statistics, sort of how music playing licensing does work in the USA too. So argument here about whether libraries really do this or not may just be confusion over the fact that things really do work differently in different countries.

I suppose that gives us the opportunity to compare... has the right to loan out legally purchased books without licenses damaged the incentive to write books in the USA compared to Europe? Like even back in pre-ebook days? I suspect not.