Migrates to Eslint's flat config (#6276)

## What's the problem this PR addresses?

This lets us remove the Rushstack fix, and dogfood the VSCode SDK with
the flat config (which is the only available config in Eslint 9). I
would have migrated us to Eslint 9, but eslint-plugin-react [isn't
compatible
yet](jsx-eslint/eslint-plugin-react#3699).

## How did you fix it?

Migrate to eslint.config.mjs.

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Author: arcanis

.eslintignore

@@ -7,6 +7,7 @@
     "watchDirectory": "useFsEvents"
   },
   "npm.packageManager": "yarn",
+  "eslint.experimental.useFlatConfig": true,
   "eslint.validate": [
     "javascript",
     "typescript",