chore: Use '@typescript-eslint/parser' for ESLint

[smockle]

Follow-up to #959
Part of #970

The current ESLint configuration surfaces many false negatives (“before” screenshot below). By adopting configuration and tooling closer to that used in the main GitHub repo and by GitHub’s React projects (which use @typescript-eslint/parser), only valid errors remain (“after” screenshot below).

Screenshots

Before	After

Merge checklist

• [-] Added or updated TypeScript definitions (index.d.ts) if necessary
• [-] Added/updated tests
• [-] Added/updated documentation
• [-] Tested in Chrome
• [-] Tested in Firefox
• [-] Tested in Safari
• [-] Tested in Edge

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Latest commit: e1e17d3

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[bscofield]

👍 🙇

[colebemis]

Interesting, a lot of the linting errors are coming from no-shadow which is disabled in the config 🤔

[smockle]

Interesting, a lot of the linting errors are coming from no-shadow which is disabled in the config 🤔

Fixed in 2214db8. TIL rule overrides (like the existing "no-shadow": 0) must be in the same block as the extends’d ruleset they override (which I’d moved).