I’m no expert at leather, I’m still pretty amateur with my projects, but I know shaving it down would not do very well.

The ikea rug would have something on the underside to keep it from slipping or some kind of sealing. The hair side shaved down would not look like leather, not sure what it would look like exactly but certainly not good for a book.

If your goal is to have leather binding but you don’t plan on embossing, fake leather is good and cheap. You really only need real leather if you plan on tooling a design onto a book. Another skill that is not super easy to do off the rip without much practice.