FYI the Kobo Libra 2 is just this if you don't need the Amazon ecosystem. If you do, you're out of luck for now though.
I can check out a book from the library on my phone and have it appear wirelessly on my Kindle moments later. Last I checked all non-Kindle devices require you to basically download an .epub file onto your computer and then sideload it with a cable to check out a library book, which is too much hassle for something I do several times a week.
My process for checking out books on a Kobo is:
- check out book on Libby app
- go to Kobo
- hit the sync button.
The last time I tried to check out a book for Amazon (admittedly a while ago) it was - check out book on Libby app
- get redirected to Amazon & hit "borrow" <-- just a bit of extra friction
- go to Kindle
- hit the sync button