Migrate to new Dart web package

[fzyzcjy]

Changes

Close #2039

Checklist

• An issue to be fixed by this PR is listed above.
• New tests are added to ensure new features are working. Please refer to this page to see how to add a test.
• ./frb_internal precommit --mode slow (or fast) is run (it internal runs code generator, does auto formatting, etc).
• If this PR adds/changes features, documentations (in the ./website folder) are updated.
• CI is passing. Please refer to this page to see how to solve a failed CI.

Remark for PR creator

• ./frb_internal --help shows utilities for development.
• If fzyzcjy does not reply for a few days, maybe he just did not see it, so please ping him.