Revert bindings dependency workaround

[goderbauer]

DO NOT SUBMIT until customers are migrated!! Customers are migrated.

Description

This reverts commit 0089896 of #54212, which was only submitted to allow migrating our customers without breaking them.

Related Issues

Pre-work for #6827.

Tests

I added the following tests:

• adjusted tests to make them no longer dependent on the workaround.

Checklist

Before you create this PR confirm that it meets all requirements listed below by checking the relevant checkboxes ([x]). This will ensure a smooth and quick review process.

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• I signed the CLA.
• I read and followed the Flutter Style Guide, including Features we expect every widget to implement.
• I read the Tree Hygiene wiki page, which explains my responsibilities.
• I updated/added relevant documentation (doc comments with ///).
• All existing and new tests are passing.
• The analyzer (flutter analyze --flutter-repo) does not report any problems on my PR.
• I am willing to follow-up on review comments in a timely manner.

Breaking Change

Did any tests fail when you ran them? Please read Handling breaking changes.

• No, no existing tests failed, so this is not a breaking change.
• Yes, this is a breaking change. If not, delete the remainder of this section.
  • I wrote a design doc: https://flutter.dev/go/template Replace this with a link to your design doc's short link
  • I got input from the developer relations team, specifically from: Replace with the names of who gave advice
  • I wrote a migration guide: Migration guide for reversing dependency of scheduler/services website#3918

[goderbauer]

(marked WIP so it is not accidentally submitted before migration is done)

[dnfield]

LGTM once internal customers ready

[goderbauer]

Internal customers have been migrated (cl/305890946). This will submit when the tree turns green.