Migrate developer script to list-login-items-for-app cmd

[danielbayley]

Part of #174192.

• list_login_items_for_app

---

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 (add your cask's name to the end of the search field).
• brew audit --cask --new <cask> worked successfully.
• HOMEBREW_NO_INSTALL_FROM_API=1 brew install --cask <cask> worked successfully.
• brew uninstall --cask <cask> worked successfully.

---

• AI was used to generate or assist with generating this PR. Please specify below how you used AI to help you, and what steps you have taken to manually verify the changes, including zap stanza paths.

---