Update example tab on pub.dev and remove low-quality dependency

[fzyzcjy]

Changes

Close #1260
Close #1262

Checklist

• An issue to be fixed by this PR is listed above.
• New tests are added to ensure new features are working. End-to-end tests are usually in the ./frb_example/pure_dart example, more specifically, rust/src/api.rs and dart/lib/main.dart.
• The code generator is run and the code is formatted (via just precommit).
• If this PR adds/changes features, documentations (in the ./book folder) are updated.
• CI is passing.

Remark for PR creator

• Justfile is a task runner for the command line interface that allows you to run shell commands from a file named justfile in your project directory. You can use Justfile after installing it. Note that commands written in justfile of this repository are expected to be run in bash, not cmd or powershell. Running just ... commands in cmd or powershell will produce errors as the syntax is not compatible. On Windows, you can use git bash if you have Git installed.
• If fzyzcjy does not reply for a few days, maybe he just did not see it, so please ping him.

[fzyzcjy]

(would like to discuss #1261 (comment) before further implementation)