Let’s see if I understand what you are doing, because I’m not sure. You are using some not quite right terms, and I’m only half sure my guesses are right.
You are trying to read your Kindle Unlimited books in the web browser on the Amazon site. You’ve tried Apple’s Safari browser, Google’s Chrome browser, and Microsoft’s Edge browser. You can’t load new books, and can not delete (stop reading) the book you just finished.
If I’m right, I don’t really know why the book reader on the Amazon site i not working. I’ve never tried using it. Maybe Amazon has designed it poorly for tablets. The fact that it is not working on three different browsers suggests it is not just a browser glitch that can be fixed by clearing history.
My suggestion is to use the Kindle App instead. This is a far better reading experience on the iPad anyway. The description of Kindle Unlimited on the site says it will work on any device on the Kindle app; though I can’t offer details. I don’t have Kindle Unlimited myself.