test(docs): make search and nav e2e tests hydration-aware (#551)

The docs e2e suite clicked navbar and sidebar controls before Vue hydrated the
multi-locale bundle, so the clicks were swallowed. That raced the Pagefind
search open (filed as #551), the appearance toggle, and the homepage and
sidebar navigation tests. The old search tests also matched an input
placeholder the config overrides, so they failed against a working build.

Add a waitForHydration helper (gates on #app.__vue_app__, set inside Vue's
app.mount()) and an openDocsSearch helper, and route the affected tests through
them. Search itself was never broken; this change is test-only. Docs e2e suite
is green (43/43).

Closes #551

tests/e2e-docs/helpers.ts

@@ -0,0 +1,45 @@
+import { expect, type Locator, type Page } from "@playwright/test";
+
+/**
+ * Wait for VitePress to finish the initial client hydration.
+ *
+ * The docs navbar (the Pagefind search box, the appearance toggle) is
+ * server-rendered, so it's present in the DOM the instant the page loads, but
+ * its Vue click handlers aren't wired until the app hydrates. A click that
+ * lands before then is silently swallowed. That's the flaky "search won't
+ * open" race behind issue #551: the bundle for 21 locales takes a few hundred
+ * ms to hydrate, and a fast (or automated) click inside that window does
+ * nothing. Playwright's actionability checks don't cover framework hydration,
+ * so tests have to wait for it explicitly.
+ *
+ * Vue assigns `__vue_app__` to the mount container inside `app.mount()`, which
+ * for SSR is the same call that hydrates the initial tree. Its presence on
+ * `#app` is therefore a deterministic, side-effect-free signal that handlers
+ * are attached.
+ */
+export async function waitForHydration(page: Page): Promise<void> {
+  await page.waitForFunction(
+    () => {
+      const app = document.getElementById("app");
+      return app !== null && "__vue_app__" in app;
+    },
+    null,
+    { timeout: 15_000 },
+  );
+}
+
+/**
+ * Open the Pagefind search dialog and return its input locator.
+ *
+ * Waits for hydration first (so the click registers), then asserts the dialog
+ * input is visible. The selector is placeholder-agnostic on purpose: the docs
+ * config overrides the input placeholder, so keying off a hard-coded string
+ * (as the old tests did) breaks whenever that copy changes.
+ */
+export async function openDocsSearch(page: Page): Promise<Locator> {
+  await waitForHydration(page);
+  await page.locator(".nav-search-btn-wait").first().click();
+  const input = page.locator("[command-dialog-wrapper] input").first();
+  await expect(input).toBeVisible({ timeout: 5_000 });
+  return input;
+}