[NoahStapp]

PYTHON-6056 followup.

Changes in this PR

Retry up to a 10 minute timeout for transient search index management errors during the search index management tests and their cleanup. Reduce code duplication in similar tests.

Test Plan

Existing test changes.

Checklist

Checklist for Author

• Did you update the changelog (if necessary)?
• Is there test coverage?
• Is any followup work tracked in a JIRA ticket? If so, add link(s).

Checklist for Reviewer

• Does the title of the PR reference a JIRA Ticket?
• Do you fully understand the implementation? (Would you be comfortable explaining how this code works to someone else?)
• Is all relevant documentation (README or docstring) updated?

[Copilot]

🟢 Approval recommended

No active review issues were identified for the search-index polling and retry changes.

No open issues 🎉

What changed in this PR

This PR reduces flakiness in PyMongo’s search index management tests by centralizing polling and retrying transient search service connection errors.

Changes:

• Adds a shared polling helper with a 10-minute timeout and five-second retry interval.
• Uses the helper for index readiness, drop requests, and deletion checks.
• Removes duplicated deletion polling from the prose test.

File	Description
test/​test_index_management.py	Mirrors the retry and polling changes in the generated synchronous tests.
test/​asynchronous/​test_index_management.py	Adds asynchronous retry polling and reuses it for readiness and deletion checks.

_{Review effort: Balanced}