Lots of libraries use the Libby app to reserve and “borrow” ebooks, and it does indeed give them to your kindle library for the loan period. Your specific system may be different, but I’d check it out.

I also highly recommend libby, which lets you check out ebooks and audiobooks from your library. I don't have a kindle myself, but this help article says it's supported "Reading Kindle Books on a Kindle ereader"

You can also add multiple library cards, so if you wanted to go crazy you can find libraries that let you sign up for a card even if you don't have a local address and get access to both library's collections to read on your Kindle

Libby is 100% compatible with kindle, has been for years. It's the first prompt immediately after you check out the book; it asks if you want to read it in Libby, or have it sent to your kindle. I'd definitely recommend it, it couldn't be more seamless.