[flutter_plugin_tools] Migrate publish-check to the new base command

[stuartmorgan-g]

To support this command's --machine flag, which moves all normal output into a field in a JSON struct, adds a way of capturing output and providing it to the command subclass on completion.

Part of flutter/flutter#83413

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 relevant style guides and ran the auto-formatter. (Note that unlike the flutter/flutter repo, the flutter/plugins repo does use dart format.)
• I signed the CLA.
• The title of the PR starts with the name of the plugin surrounded by square brackets, e.g. [shared_preferences]
• I listed at least one issue that this PR fixes in the description above.
• I updated pubspec.yaml with an appropriate new version according to the pub versioning philosophy.
• I updated CHANGELOG.md to add a description of the change.
• 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.