If you’re in a country with a PLR scheme, there actually is a payment per rental.

Look who hasn’t heard of PLR lol

I don’t have a lot of knowledge, but the ELR/PLR system that pays me thousands of dollars a year from library reads is funded by the federal government. The public libraries themselves are funded by local councils (80%) and state governments (20%), and the size of the population in the administrative area determines the allocation of funds.

This is specific to my state (Victoria, Australia) but may answer more of your questions.

DOLLARS, SENSE AND PUBLIC LIBRARIES https://www.slv.vic.gov.au/sites/default/files/Dollars-sense-public-libraries-report.pdf

Australian author here! We have public lending right payments from libraries, and educational lending right payments from school libraries. For me, this has already totalled about amount exceeding the original advance for my first book. They can keep the money coming in for years after book goes out of print.

Libraries support authors, directly and indirectly. Books get bought (sometimes in multiples), promoted, and replaced in libraries. People buy books themselves from authors they got acquainted with in libraries. Some libraries pay us directly for events we run with them. Libraries are absolutely not the same as piracy. Use libraries.