Migrate test coverage from test_everest_client.py

Asking copilot on what test functionality that was lost in #14272 gave this response. This issue is to address the gap.

> Gap: The specific negative-path behaviors of stopping an experiment — non-200 response, connection-down, 404/missing-endpoint — are no longer tested anywhere.  stop_experiment_server()  also changed behavior (silent retry loop returning a bool, no longer logging distinct error messages), and that new retry logic itself is untested. The full concurrent-multi-client integration scenario was also downgraded to a narrower server-side unit test rather than a true end-to-end equivalent.
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> Bottom line: Most of the removed file's intent is covered by relocated/adapted tests, but the error-handling paths of stopping the server, and a true multi-client end-to-end integration check, have no replacement test today.

Inspect the current tests that replaced test_everest_client.py and the stop_experiment_server() implementation; run those tests first. Done means covering non-200, connection-down, 404/missing-endpoint, retry behavior, and a true concurrent multi-client end-to-end scenario.

Written by the indexing model from the issue text.