I have NEVER used the Kindle experimental browser to access my public library site. I don't think it would work, the browser is basically useless.
I use my iPad browser, or the computer if necessary.
For normal buying or using Kindle Unlimited, I rarely use the Kindle. I do it from an iPad or computer.
Kindle doesn't have the same type of direct link that the Kobo Aura One has to public libraries. But, once a book is borrowed, it is available for any Kindle or Kindle app on my Amazon account.
None of my comments pertain to Amazon's Fire tablets. I don't use them, but assume their browser would be fine for accessing public libraries.
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