Fix race condition in overall_experience_test

[Hixie]

Fixes #75860

Pre-launch Checklist

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• I read the Tree Hygiene wiki page, which explains my responsibilities.
• I read and followed the Flutter Style Guide, including Features we expect every widget to implement.
• I signed the CLA.
• I listed at least one issue that this PR fixes in the description above.
• I updated/added relevant documentation (doc comments with ///).
• I added new tests to check the change I am making or feature I am adding, or Hixie said the PR is test-exempt.
• All existing and new tests are passing.

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[jonahwilliams]

LGTM