I think you're thinking along the right lines, but I am skeptical that this protocol adds much. A saved payment request is meant to be unique per transaction e.g. because the destination address is unique for that payment (for privacy reasons). Where would you store the signed payment request? Probably in the wallet. You could just extract the metadata that's useful for UI rendering into a separate structure and then encrypt the original full payment request under the wallet key. At least this is how I imagine it would work.

So then, if someone can steal a payment request they can probably steal the wallet signing keys too, and thus signing a challenge with the wallet keys doesn't add much. It means the wallet doesn't have to store the PaymentRequest encrypted. But AFAICT that's about all it does.