Move from internal expect rule to eslint-plugin-expect-type

#843 migrated the old TSLint Expect rule to an ESLint rule (🥳!). Great!

There exists a community plugin for roughly the same area of functionality already: https://github.com/JoshuaKGoldberg/eslint-plugin-expect-type [^1]. It would be great if we could use the standard plugin instead. That way bugfixes + features from DT usage can help other community members, and vice versa.

Proposal:

1. For each bug or feature that is better in this repo, we can file a corresponding issue on eslint-plugin-expect-type
2. Once those are all fixed up, move to using the external plugin

Per later comments, tracking issues in eslint-plugin-expect-type for missing features:

• https://github.com/JoshuaKGoldberg/eslint-plugin-expect-type/issues/107
• https://github.com/JoshuaKGoldberg/eslint-plugin-expect-type/issues/108
• https://github.com/JoshuaKGoldberg/eslint-plugin-expect-type/issues/109

[^1]: I do feel a bit weird promoting a package with my name on it. If it helps, I didn't create it - just took over maintenance when the previous maintainer didn't have time. If there's a better alternative to eslint-plugin-expect-type that'd be great too!

Review the current internal Expect rule and ESLint setup, then check the three linked eslint-plugin-expect-type tracking issues (#107–#109). Done means those missing features are resolved and this repository uses the external plugin instead.