Authors get paid for library books. (They often cost quite a bit more than the same book sold to an individual.) An author/publisher can choose not to sell to a particular library if they really want to.

They do, but publishers have been tightening the screws on libraries too.

Libraries pay the ordinary retail price (or less from a bulk discount). They sometimes buy books with (more expensive) sturdier binding, expecting it to be used more heavily.

Authors cannot stop libraries from buying their books and putting them on the shelf.

Or are you talking about ebooks?

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.

Libraries pay more for "library binding" copies of books, when available

They may choose to, because they are willing to pay more for the longer-lasting binding. Or it may help them remain on good terms with publishers. While I'm technically a librarian, I haven't worked in this area or at an institution where it's relevant for a long time now, so actually don't know personally how common this is.

But if libraries choose to pay more for a better binding, that is because of their determination that the value is worth it to them, not because there is any law that says they must to give more compensation to copyright holders. Nor is it proportionate to how many times they lend out the book or anything.

In the USA libraries are completely legally allowed to take a copy they bought legally anywhere at all, including a used booksale from someone else, and loan it out as many times as they want, without needing a license or to make any additional payments to original copyright holder. I am so so positive of this (although I am not a lawyer).

The current legal issue around the Internet Archive's library came to a head because IA decided to remove restrictions on lending. They changed from the "we have one copy, so we lend one copy" model that libraries follow to "we have a one copy, so everyone can have a copy at the same time" for a period of time.

Had they not done that, the publishers knew they faced an uphill battle because of the first-sale doctrine. But once Pandora's box was opened, the publishers saw their chance to stamp out the whole thing by arguing "well they did it once, what's to stop them from doing it again?"

IA made a well-intentioned but ill-advised decision.