Fix admin-Notice to not render into REST requests, etc.

[carstingaxion]

Description of the Change

Closes # Follow-up Bug fix for #739

How to test the Change

Changelog Entry

Fixed - Fix alpha-admin-notice to not render into REST requests, etc.

Credits

Props @carstingaxion

Checklist:

• I agree to follow this project's Code of Conduct.
• I have updated the documentation accordingly.
• I have added tests to cover my change.
• All new and existing tests pass.

[what-the-diff]

PR Summary

• Updated Method Trigger in Class Setup
  The technical action that triggers the check_gatherpress_alpha method within the class-setup.php file has been updated. With this change, the method now activates in response to administrative notifications rather than at initial administration load. This adjustment ensures that the function executes at the right time when admin notifications are being dealt with, increasing overall efficiency.