Except this is exactly how Android Pay and Apple Pay work: they store the billing information on the client side. A large part of the impetus behind Payment Requests is to open up Android Pay and Apple Pay to the web. And all(?) major browsers support autofilling card details already, which means storing the billing information on the client side and exposing the web to that fraud risk.
And, heck, a conforming Payment Requests payment handler doesn't need to store anything locally: one based on payment cards could require you to enter the card details on every transaction.