Fixes deprecations causing unexpected standard error integration test failures: https://github.com/flutter/flutter/issues/138061, https://github.com/flutter/flutter/issues/138063, https://github.com/flutter/flutter/issues/138067, https://github.com/flutter/flutter/pull/138077.

Note that the issue this fixes was the same that was the cause for reverting https://github.com/flutter/flutter/pull/137191. I made this same fix in https://github.com/flutter/flutter/pull/137967 and changed `Mac_android run_release_test` to run in presubmit to see a successful run for proof that this fix works.