To quote from the article you linked to (emphasis mine):
The Payment Request API is designed to be vendor-agnostic, meaning it does not require use of a particular payment system. It's not a new payment method, nor does it integrate directly with payment processors; rather, it is a conduit from the user's payment and shipping information to merchants, [...]
In other words, the purpose of this API is to allow you to receive payment information, e.g. a payment card number or some sort of payment token. It's not an end-to-end solution for payments - just an easier and safer approach to the checkout experience.
Furthermore, the Payment Request API is an open standard - it's not tied to Google and different browsers may have different implementations. Your second link (list of 3rd party payment processors) actually refers to Google Pay, which is a Google product unrelated to the Payment Request API (however, Google Pay can be used as a payment method through Payment Request API on supported browsers).