Depending on the implementation, you can run the app on your computer. I don't see why the iOS app wouldn't work on macOS, and there are tons of tools to run Android apps on Windows and Linux.

If the actual implementations do copy the dependency on Play Integrity and other such APIs, that does become a problem (getting past that is a major annoyance on amd64 computers because there are so few real amd64 Android devices that can be spoofed).

However, the law regarding these apps specifically states that the use of this app must be optional. I'm not sure websites and services will implement other solutions, but in theory you should not need a phone unless you want the convenience and privacy factor of app verification. I expect alternatives (such as 1 cent payments with credit cards in your name) to stick around, at least until we get a better idea about how this thing will work out in practice.