> Who does the selection?

Of what?

> Who approves the applications?

What's to "approve"?

> Who ensures the process is open?

Put it on a website and in the local paper. Applicants' names, their exam scores, their law school transcripts.

> It always comes down to winners of elections

Sure somewhere in the chain, at the end, there's an elected official. That's democracy.

But elected officials can be required by law to use meritocratic selection criteria to appoint people. Many countries used or still use exams to hire bureaucrats. It's a very old concept:

https://en.wikipedia.org/wiki/Civil_service_entrance_examina...

https://en.wikipedia.org/wiki/Imperial_examination

> The USA isn’t a technocracy or meritocracy, for better or worse.

That's fine but a different argument from "it can't be done".