chore: remove Playwright end-to-end testing setup

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

playwright.config.ts

@@ -1,20 +0,0 @@
-import { defineConfig, devices } from "@playwright/test";
-
-const baseURL = process.env.PLAYWRIGHT_BASE_URL ?? "http://127.0.0.1:3000";
-
-export default defineConfig({
-	testDir: "./e2e",
-	fullyParallel: true,
-	retries: process.env.CI ? 2 : 0,
-	reporter: process.env.CI ? "github" : "list",
-	use: { baseURL, trace: "on-first-retry" },
-	webServer: process.env.PLAYWRIGHT_BASE_URL
-		? undefined
-		: {
-				command: "pnpm dev --port 3000",
-				url: "http://127.0.0.1:3000/api/health",
-				reuseExistingServer: !process.env.CI,
-				timeout: 120_000,
-			},
-	projects: [{ name: "chromium", use: { ...devices["Desktop Chrome"] } }],
-});