To be clear, my 'not fully accurate' was in response to your "Local searches do not remain private". Local searches do remain private, and there are several ways of doing them. But the default search is not a local-only search.

I agree with you: I expect the default search to be local-only, and I've turned off remote results. But it's not as though it hides the fact that it's searching remotely. So I stand by my words: it's not fully accurate to say that local searches are exposed.