Open router supports per-request usage metering, as well as per-key. Building user-level metering from this is trivial, and there are many implementations of this out there. Supporting per-user first-class is trivial considering Open router already has per-request and already supports per-request attribution. How important that is, is a big question, but definitely not something that requires Stripe to build.

Could there be use cases where you want to run inference for users in your app, but don't want to host your own backend, user authentication, and billing?

The user could log into his personal OpenRouter account, authorize your application, and optionally set a budget, all in something like a Stripe payment screen.