Libby (and by extension Overdrive) supports checking out books onto supported e-readers. They must support either the Kindle or Adobe DRM standards though.
I use overdrive from my kobo. I have three library cards in it, unfortunately I can't use them three at the same time so depending on what kind of book I want, I switch accounts. I can search, reserve, and borrow from any of those consortiums directly from the Kobo device.
Isn't this what Libby - https://libbyapp.com/ - does?
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