I agree that the signature for process_handler is weird, but you could steelman it to take a borrowed trait object instead, which would have an extra sigil.
The handler function isn't actually unnecessary, or at least, it isn't superfluous: by default, the signature would include 'a on self as well, and that's probably not what you actually want.
I do think that the example basically boils down to the lifetime syntax though, and yes, while it's a bit odd at first, every other thing that was tried was worse.