chore: update livechecks for GitHub API usage

[SMillerDev]

Important: Do not tick a checkbox if you haven’t performed its action. Honesty is indispensable for a smooth review process.

In the following questions <cask> is the token of the cask you're submitting.

After making any changes to a cask, existing or new, verify:

• The submission is for a stable version or documented exception.
• brew audit --cask --online <cask> is error-free.
• brew style --fix <cask> reports no offenses.

Additionally, if adding a new cask:

• Named the cask according to the token reference.
• Checked the cask was not already refused.
• Checked the cask is submitted to the correct repo.
• brew audit --new-cask <cask> worked successfully.
• brew install --cask <cask> worked successfully.
• brew uninstall --cask <cask> worked successfully.

[SMillerDev]

See also Homebrew/homebrew-core#130233 and Homebrew/brew#15270

[razvanazamfirei]

Ran brew style --fix. Hopefully it will get rid of the syntax failure.

[samford]

I took care of appium (#146747), brook (#146749), and ricochet (#146754) in separate PRs, since they required additional changes.

Razvan beat me to rebasing this (dropping the commits for brook and ricochet) but I had to rebase it again to drop the appium commit. I'm currently doing a test run to make sure this takes care of all the existing breakages at this point.

Edit: The test run was good, so this can be merged when ready.

[razvanazamfirei]

I'm just happy to be in y'all's company!

[samford]

As mentioned in Homebrew/homebrew-core#130233 (review), I'll create follow-up PRs to update the regexes to use ^/$ (like we do with Git regexes), so we can get these fixes merged in the interim time.