chore(lint): apply the main eslint config to vue files (#39545)

Vue files only got `eslint-plugin-vue` rules, so stylistic and
TypeScript rules never ran on them, and `import-x` could not see cycles
between `.ts` and `.vue` files.

- Apply the main ESLint config to `.vue` files
- Let `import-x` parse `.vue` files, which surfaced a cycle between
`repo-findfile.ts` and `RepoFileSearch.vue`
- Fix the resulting lint issues

`vue-eslint-parser` is added as a direct dependency because `import-x`
resolves parsers by package name, see
https://github.com/un-ts/eslint-plugin-import-x/issues/381.

package.json

@@ -109,6 +109,7 @@
     "typescript": "6.0.3",
     "typescript-eslint": "8.70.1",
     "vitest": "5.0.1",
+    "vue-eslint-parser": "10.4.0",
     "vue-tsc": "3.3.11"
   }
 }