Use aiohasupervisor for store APIs

[mdegat01]

Proposed change

Continuing off recent PRs in the supervisor integration, migrate all calls to Supervisor's store API to run through aiohasupervisor.

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[mdegat01]

For reviewers - the increase in call count here is not actually a change. If you look in #125926 you'll see that I decreased each of these numbers by 2 to reflect the fact that mocking had changed for 2 calls we were previously making. Those 2 calls were going through the mocked client library as opposed to the mocked websession.

I realized in this PR that a better way to change these tests was to add the mock calls to these two things together to show that the call count truly had not changed. So as a result I had to increase these numbers back to what they were before the prior PR.