Migrate to eslint-plugin-matrix-org

This migrates to the new plugin form of our custom ESLint configs. As part of
this, some packages are de-duplicated, configs streamlined, etc.

package.json

@@ -71,6 +71,8 @@
   },
   "devDependencies": {
     "@babel/core": "^7.12.10",
+    "@babel/eslint-parser": "^7.12.10",
+    "@babel/eslint-plugin": "^7.12.10",
     "@babel/plugin-proposal-class-properties": "^7.12.1",
     "@babel/plugin-proposal-decorators": "^7.12.12",
     "@babel/plugin-proposal-export-default-from": "^7.12.1",
@@ -92,8 +94,9 @@
     "@types/react": "^17.0.2",
     "@types/react-dom": "^17.0.2",
     "@types/sanitize-html": "^2.3.1",
+    "@typescript-eslint/eslint-plugin": "^4.17.0",
+    "@typescript-eslint/parser": "^4.17.0",
     "autoprefixer": "^9.8.6",
-    "babel-eslint": "^10.1.0",
     "babel-jest": "^26.6.3",
     "babel-loader": "^8.2.2",
     "canvas": "^2.6.1",
@@ -102,8 +105,8 @@
     "cpx": "^1.5.0",
     "css-loader": "^3.6.0",
     "eslint": "7.18.0",
-    "eslint-config-matrix-org": "^0.2.0",
-    "eslint-plugin-babel": "^5.3.1",
+    "eslint-config-google": "^0.14.0",
+    "eslint-plugin-matrix-org": "github:matrix-org/eslint-plugin-matrix-org#main",
     "eslint-plugin-react": "^7.22.0",
     "eslint-plugin-react-hooks": "^4.2.0",
     "extract-text-webpack-plugin": "^4.0.0-beta.0",