Yarn is currently being powered by a service running at https://registry.yarnpkg.com - at least that's what's referenced in the yarn.lock file that's created.

I see no documentation nor code for the service powering this, nor is there a way to tell the command line tool to use a registry at a different address.

Aren't we just replacing the dependency on npm, Inc.'s registry with a new dependency on Facebook Inc's registry with this?

Yes. The client has a few advantages, but the registry is still closed and while with npm there's an option of running a private registry if you pay them, here, there's no option at all.