[FEAT]: Update `_reusable-check-api-for-breaking-changes.yml` to take advantage of new public griffe features

[nfelt14]

Description

There are some new, public griffe features that should be used to make the output more user-friendly.

1. https://mkdocstrings.github.io/griffe/guide/users/checking/#markdown
2. https://mkdocstrings.github.io/griffe/guide/users/checking/#github

Additional Information

No response

[nfelt14]

The markdown may not be as useful, it seems to be more adapted for changelogs, but it would still be worth it to look into.

[nfelt14]

Released in v1.6.0