feat(players): keep a video playing in the background on Brave iOS; repeat on the watch page

On iPad Brave, a shared video stopped the moment the browser left the
foreground, even with Brave's "Enable Background Audio" on, while an mp3
kept playing. Brave iOS keeps a web <video> going by re-playing one the OS
paused (brave-core ios/browser/web/media/resources/media_backgrounding.ts),
but it only hooks a <video> present at document start or added as a
MutationObserver addedNode that IS a video. React inserts a player's
<video> inside its freshly mounted wrapper in one go, so Brave never saw it.

lib/browserBackgroundPlay re-inserts a mounted <video> at its own position,
once per element: it becomes an added node, the DOM React manages does not
change, and playback is not interrupted. The watch page, the downloads
library player and the Plex player hand their <video> to it. Measured on the
owner's iPad (Brave) with a prototype of the same idea: the video kept
playing in another app and with the screen locked. The YouTube modal is not
ours to change: its <video> lives in YouTube's cross-origin iframe, and
nothing of ours pauses it on backgrounding.

The prototype's two audio-element variants (a muted video following an
<audio>, and an <audio> taking over on hide) went silent in the background on
the iPad and are removed.

The watch page gets a Repeat toggle, on by default and remembered per
browser (LS.watchRepeat; the page has no account): it sets `loop` on the
<video> or the audio-only <audio>.

frontend/src/components/LibraryPlayer.dom.test.tsx

@@ -11,6 +11,8 @@ vi.mock("react-i18next", async (importOriginal) => {
   const actual = await importOriginal<typeof import("react-i18next")>();
   return { ...actual, useTranslation: () => ({ t: (key: string) => key }) };
 });
+const expose = vi.hoisted(() => vi.fn());
+vi.mock("../lib/browserBackgroundPlay", () => ({ exposeVideoToBackgroundPlay: expose }));
 
 import LibraryPlayer from "./LibraryPlayer";
 import type { DownloadJob } from "../lib/api";
@@ -90,6 +92,11 @@ describe("LibraryPlayer", () => {
     expect(document.body.querySelector("audio")).toBeNull();
   });
 
+  it("hands its <video> to the background-play hook, so Brave iOS keeps it playing", () => {
+    render(<LibraryPlayer job={withAsset({ width: 1280, height: 720 })} onClose={() => {}} />);
+    expect(expose).toHaveBeenCalledWith(document.body.querySelector("video"));
+  });
+
   it("plays an audio-only asset (media_kind='audio') in an <audio> element", () => {
     render(
       <LibraryPlayer