This PR makes the following changes:

1. Remove dependency on [peters/evans package](https://github.com/marketplace/actions/create-pull-request)<br>
The market place action introduces overheads that don't properly consume tokens. e.g. :[failed workflow that says token is not supplied](https://github.com/flutter/flutter/actions/runs/7282529195/job/19845096943)
This PR changes the market place action to git commands that we have full control over, provides better error msg for debug, and properly consumes token.

2. Align usage of tokens:<br>
All tokens in the workflow now uses flutter actions bot PAT token. From experiments, a mixed usage of different tokens in different steps sometimes cause the workflow to fail on authentication.

Tested: Tested with [a similar workflow on my personal repository](https://github.com/XilaiZhang/miscellaneous-side-project/blob/master/.github/workflows/easy-cp.yml), and it produces the [expected cherry pick PR as end result](https://github.com/flutter/flutter/pull/140497)