feat(ci): revamp formatting workflow

[novusnota]

Closes #1006. Let's start anew.
Towards #1352 — The workflow responds to commands within comments, but doesn't post new comments yet.

TODO:

• /fmt
• autofixes for PRs on review
• miscellaneous enhancements of the formatter and its comments

Prior notes

Things to try:

1. Different event activity triggers and check statuses
2. Merged version of this PR and such checks (in a repo for experiments)
3. Creating a "GitHub App" and placing it here in the root -- that has the least chances of success because it requires listening to webhooks and I don't want to spawn extra infra for CI right now.

UPD: After some testing, I know that:

1. Different event activities did not help
2. check_run always runs from the base (like pull_request_target), so merge was required. That said, it did not help either
3. Buttons in the UI can only be displayed via GitHub Apps — we won't do that