Generate test metrics consistently.

[godofredoc]

Starts creating metric files consistently for web, dart and flutter runners. It also starts creating the files in the temp folder instead of the checkout one.

Bug: https://github.com/flutter/flutter/issues/145793

Pre-launch Checklist

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

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