Try to convince every website owner ever to accept the same payment methods.

Both Brave and Flattr are attempting the same, both have little to no success for that simple reason.

To be fair those are not major browsers. Firefox would be an obvious experimental vehicle, and the choice of bitcoin would be the least objectionable payment method (of the alternatives).

Web Payments API is on the way.

You can have many Bitcoin wallets. And yes, at least one online, to pay stuff from. But you don't keep substantial balances in them.