[integration_test] Reland add a run method for proper reporting of test results
#70513
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.

Description
Reland #70075
See 2a3256e for the changes made against the previous PR. These changes should already be present on HEAD in
binding_fail_test.dart, but somehow Git isn't able to detect that the file has moved.Also ran
..\..\bin\flutter test(from theintegration_testpackage on my windows machine which passesRelated Issues
Fixes #50816
Fixes #70467