ci: add type-checking for eslint v10

[michaelfaith]

PR Checklist

• Addresses an existing open issue: fixes 🚀 Feature: ESLint v10 support #412
• That issue was marked as status: accepting prs
• Steps in CONTRIBUTING.md were taken

Overview

This change updates the version of eslint in our codebase to v10, and adds additional type-checking in ci, running that job against both v9 and v10.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 100.00%. Comparing base (3422fe4) to head (9383b6e).

Additional details and impacted files

@@            Coverage Diff            @@
##              main      #413   +/-   ##
=========================================
  Coverage   100.00%   100.00%           
=========================================
  Files            6         6           
  Lines           46        46           
  Branches        11        11           
=========================================
  Hits            46        46           

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[michaelfaith]

@JoshuaKGoldberg similar to JoshuaKGoldberg/eslint-plugin-package-json#1554 i've added an ESLint version matrix with 9 and 10 to the type-check job to ensure we're not breaking v9 compatibility, as we work in v10. So, the branch ruleset will need updating again.

Ultimately, this is blocked by typescript-eslint/typescript-eslint#11952 though, so we can do linting in our repo with v10.