brew: CI experiences failures due to Homebrew being removed from $PATH on GitHub runners

brew config output

N/A

brew doctor output

N/A

Verification

• I ran brew update and am still able to reproduce my issue.
• I have resolved all warnings from brew doctor and that did not fix my problem.

What were you trying to do (and why)?

CI that uses the 20220922.2 Ubuntu image fails due to GitHub removing Homebrew from $PATH.

https://github.com/actions/runner-images/issues/6283

What happened (include all command output)?

CI fails if it uses the 20220922.2 Ubuntu runner image.

What did you expect to happen?

CI succeeds or fails

Step-by-step reproduction instructions (by running brew commands)

Example: https://github.com/Homebrew/homebrew-cask/pull/132241