🚀 Feature: Use get-github-auth-token for auth

[JoshuaKGoldberg]

Bug Report Checklist

• I have tried restarting my IDE and the issue persists.
• I have pulled the latest main branch of the repository.
• I have searched for related issues and found none that matched my issue.

Overview

I made this package a couple years ago to manually check process.env.GH_TOKEN for an auth token. I later created https://github.com/JoshuaKGoldberg/get-github-auth-token, which has nicer handling for detecting auth tokens. Accepting PRs to switch this repo to use getGitHubAuthToken!

Additional Info

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