chore: remove Playwright end-to-end testing setup

Remove Playwright config, e2e spec, dependencies, and related references from package.json, tsconfig.json, and .gitignore.

tsconfig.json

@@ -34,5 +34,5 @@
 		".next-staging/types/**/*.ts",
 		".next-staging/dev/types/**/*.ts"
 	],
-	"exclude": ["node_modules", "e2e", "playwright.config.ts", "src/generated"]
+	"exclude": ["node_modules", "src/generated"]
 }