📝 Documentation: Switch from classic GitHub personal access tokens (PATs) to fine-grained

### Bug Report Checklist

- [x] I have pulled the latest `main` branch of the repository.
- [x] I have [searched for related issues](https://github.com/JoshuaKGoldberg/create-typescript-app/issues?q=is%3Aissue) and found none that matched my issue.

### Overview

https://github.blog/security/application-security/introducing-fine-grained-personal-access-tokens-for-github: _fine-grained_ personal access tokens (PATs) are what GitHub recommends now using for security. CTA's docs still refer to https://github.com/settings/tokens/new, the page for creating new _classic_ PATs.

Now that fine-grained PATs don't have forced expiry (https://github.com/orgs/community/discussions/36441#discussioncomment-3908925), I think we're ready to update the docs to ask users to use them.

### Additional Info

I think it'd make sense to resolve this and #1045 in the same PR. Switching to the new fine-grained tokens means documenting the specific needed permissions anyway.

🎁