If the site knows you and has payment credentials stored they won't need to use Payment Request API (although they still can).
Storing payment credentials differs by payment method. If you're talking about card payments then you have to deal with things like PCI-DSS and/or tokenization but there are other ways to pay which may support this use case more explicitly, for example by capturing explicit permission from the user to allow the merchant to make future purchases seamlessly.
This is something we're trying to find a standard protocol for with https://openpayments.dev so that the ecosystem is less fragmented. But, this is not linked to the W3C work explicitly
If this API gives a fast checkout experience without storing data, then you've potentially got the benefit of storing payment info without the exposure of storing sensitive data.
(disclosure: I work at google on the web, but not on anything payments related)
At a certain scale, any reduction of friction results in $xx million in increased sales.
I ask this as someone who works at a large online retailer whose CISO has specifically asked to keep them informed of things that could improve our security posture around payments. Inability for a site to store payment information would discourage large sites from supporting this, or at best slow the implementation and support of it.
Apologies if this is somewhere in one of the two documents. I tried to skim through the ~150 pages to find related items.