One can see this as similar to %s or f-strings, where the parameter is implicitly converted to str. OTOH I'm reluctant to officially broaden the type to Any. This would be very general, and IHMO would loose the semantic aspect of the type definition.
The automatic str conversion decision may be intentional and for convenience. Due to compatibility, we're bound to the behavior whether we want or not. But I'm inclined to regard this rather as a gone-public implementation detail that we have to support, and not as an intended feature that we want to advertise.

Therefore, at least for the docstrings, I'm very much in favor of keepingstr. It may be a technical solution to make a type aliasStrLike = Any for annotations in this case, including a description why we use that. That'd be a somewhat simpler solution than thepropsedSupportsStr protocol. It would placate the type system for current code, but still carry the semantics.