chore(typecheck): replace tsc with tsrs as the typecheck gate (#8428)

tsrs (Rust port of the TypeScript 7 checker) replaces tsc --noEmit in
the typecheck script, web:build, and CI. Measured on this repository:
~0.4s vs ~3.5s wall time and ~280 MB vs ~750 MB peak RSS, with
identical zero-diagnostic results on the current tree. The typescript
package stays installed: typescript-eslint projectService and
scripts/cleanup-unused-i18n.mjs consume its compiler API.

package.json

@@ -14,7 +14,7 @@
     "build:fast": "cross-env NODE_OPTIONS='--max-old-space-size=4096' tauri build -- --profile fast-release",
     "tauri": "tauri",
     "web:dev": "vite",
-    "web:build": "tsc --noEmit && vite build",
+    "web:build": "tsrs -p . --noEmit && vite build",
     "web:serve": "vite preview",
     "prebuild": "node scripts/prebuild.mjs",
     "updater": "node scripts/updater.mjs",
@@ -32,7 +32,7 @@
     "i18n:types": "node scripts/generate-i18n-keys.mjs",
     "gen:contract": "node scripts/gen-contract.mjs",
     "arch:check": "node scripts/check-architecture.mjs listen emit invoke",
-    "typecheck": "tsc --noEmit",
+    "typecheck": "tsrs -p . --noEmit",
     "test": "vitest run",
     "test:watch": "vitest",
     "knip:check": "knip",
@@ -128,6 +128,7 @@
     "sass": "^1.104.1",
     "tar": "^7.5.22",
     "terser": "^5.51.2",
+    "tsrs": "0.20.0-ts7.1.0-beta",
     "typescript": "^6.0.3",
     "typescript-eslint": "^8.70.0",
     "vite": "^8.3.0",