If we take this to the logical extreme - someone had to buy the book in order to upload it to Anna's archive in the first place.

"If we take this to the logical extreme "

I think this is a situation where doing so doesn't make much sense. This is all about compromising, I think that must be the premise.

Yes, but whereas libraries need to buy more copies of books that lots of people check out, Anna's archive only ever needs one. Not exactly sustainable for the author.

As I said, I loved the Internet Archive's approach to this! That's very much not what Anna's archive is doing.

Still, libraries buy what, maybe 5 copies of a mildly popular book. I don't think that would be sustainable either if that was the only books sold.

Libraries have to replace paperback books after ~20 checkouts on average. (This number is from memory but I'm quite sure it's in this range.) Hardcover books last a bit longer but of course are also more expensive.

I agree the industry would have a hard time surviving off library sales alone, in the same way that most businesses rely on multiple revenue streams to make ends meet, but I think library revenue is much more significant than you're making it out to be!

It's also likely true that a library that bought 10 copies of a book initially is unlikely to buy 10 more copies once there have been 200 circulations and they are needing to be replaced. They may only buy 5 replacement copies since the book is likely to be less popular than at initial release so it will take much longer for the next 100 circulations to occur.

As for anecdota, I have more than once borrowed a library book and then purchased a copy so I could read it again or to finish it if demand is strong enough that I would have to wait weeks or months to be able to borrow it again.

The principle of virtual libraries is the same as physical ones: only one person has access to the book at any given time. For popular books, either the library has to buy more copies (or digital licenses) or else it rations access by waiting list. The idea is sound IMO.

Have you tried borrowing a mildly popular recent book from the library? There's often a digital queue of 20+ people with reservations.

There's plenty of incentive for most people to buy the real book rather than wait for the queue.

(I've also found libraries a useful way to discover lesser-known authors, since you can quickly sample/browse books on the shelves. But they wont have all of the books published by those unknown authors.... so I end up buying/ordering other things by them)

At this point, we are just arguing about what exactly the numbers are, though. There is not a black and white difference between public libraries relationship to publishers, and gray libraries relationship to publishers.