Fix occasional "Re-starting finished log context" from keyring (#8398)

* Fix test_verify_json_objects_for_server_awaits_previous_requests

It turns out that this wasn't really testing what it thought it was testing
(in particular, `check_context` was turning failures into success, which was
making the tests pass even though it wasn't clear they should have been.

It was also somewhat overcomplex - we can test what it was trying to test
without mocking out perspectives servers.

* Fix warnings about finished logcontexts in the keyring

We need to make sure that we finish the key fetching magic before we run the
verifying code, to ensure that we don't mess up our logcontexts.