This reverts commit f8f43873.

Because this class has some subtle behaviour with regards to control of
exit timing and when and how it streams data to stderr and stdout, it's
worth adding unit tests for this class directly, as well as (in a
followup patch) for FakeProcessManager.

This is additional testing relating to refactoring landed in:
https://github.com/flutter/flutter/pull/103947

Issue: https://github.com/flutter/flutter/issues/102451