feat(browser): use persistent Playwright connections for remote profile tab operations

For remote CDP profiles (e.g., Browserless), tab operations now use Playwright's
persistent connection instead of stateless HTTP requests. This ensures tabs
persist across operations rather than being terminated after each request.

Changes:
- pw-session.ts: Add listPagesViaPlaywright, createPageViaPlaywright, and
  closePageByTargetIdViaPlaywright functions using the cached Playwright connection
- pw-ai.ts: Export new functions for dynamic import
- server-context.ts: For remote profiles (!cdpIsLoopback), use Playwright-based
  tab operations; local profiles continue using HTTP endpoints
- server-context.ts: Fix ensureTabAvailable to not require wsUrl for remote
  profiles since Playwright accesses pages directly

This is a follow-up to #895 which added authentication support for remote CDP
profiles. The original PR description mentioned switching to persistent Playwright
connections for tab operations, but only the auth changes were merged.

src/browser/pw-session.ts

@@ -389,3 +389,106 @@ export async function closePlaywrightBrowserConnection(): Promise<void> {
   if (!cur) return;
   await cur.browser.close().catch(() => {});
 }
+
+/**
+ * List all pages/tabs from the persistent Playwright connection.
+ * Used for remote profiles where HTTP-based /json/list is ephemeral.
+ */
+export async function listPagesViaPlaywright(opts: {
+  cdpUrl: string;
+}): Promise<
+  Array<{
+    targetId: string;
+    title: string;
+    url: string;
+    type: string;
+  }>
+> {
+  const { browser } = await connectBrowser(opts.cdpUrl);
+  const pages = await getAllPages(browser);
+  const results: Array<{
+    targetId: string;
+    title: string;
+    url: string;
+    type: string;
+  }> = [];
+
+  for (const page of pages) {
+    const tid = await pageTargetId(page).catch(() => null);
+    if (tid) {
+      results.push({
+        targetId: tid,
+        title: await page.title().catch(() => ""),
+        url: page.url(),
+        type: "page",
+      });
+    }
+  }
+  return results;
+}
+
+/**
+ * Create a new page/tab using the persistent Playwright connection.
+ * Used for remote profiles where HTTP-based /json/new is ephemeral.
+ * Returns the new page's targetId and metadata.
+ */
+export async function createPageViaPlaywright(opts: {
+  cdpUrl: string;
+  url: string;
+}): Promise<{
+  targetId: string;
+  title: string;
+  url: string;
+  type: string;
+}> {
+  const { browser } = await connectBrowser(opts.cdpUrl);
+  const contexts = browser.contexts();
+  // Use the first context if available, otherwise this is a fresh connection
+  // and we need to use the default context that Browserless provides
+  let context = contexts[0];
+  if (!context) {
+    // For Browserless over CDP, there should be at least one context
+    // If not, we can try accessing pages directly from contexts
+    throw new Error("No browser context available for creating a new page");
+  }
+
+  const page = await context.newPage();
+  ensurePageState(page);
+
+  // Navigate to the URL
+  const targetUrl = opts.url.trim() || "about:blank";
+  if (targetUrl !== "about:blank") {
+    await page.goto(targetUrl, { timeout: 30_000 }).catch(() => {
+      // Navigation might fail for some URLs, but page is still created
+    });
+  }
+
+  // Get the targetId for this page
+  const tid = await pageTargetId(page).catch(() => null);
+  if (!tid) {
+    throw new Error("Failed to get targetId for new page");
+  }
+
+  return {
+    targetId: tid,
+    title: await page.title().catch(() => ""),
+    url: page.url(),
+    type: "page",
+  };
+}
+
+/**
+ * Close a page/tab by targetId using the persistent Playwright connection.
+ * Used for remote profiles where HTTP-based /json/close is ephemeral.
+ */
+export async function closePageByTargetIdViaPlaywright(opts: {
+  cdpUrl: string;
+  targetId: string;
+}): Promise<void> {
+  const { browser } = await connectBrowser(opts.cdpUrl);
+  const page = await findPageByTargetId(browser, opts.targetId);
+  if (!page) {
+    throw new Error("tab not found");
+  }
+  await page.close();
+}