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/lib/browserBackgroundPlay.ts

@@ -0,0 +1,23 @@
+// Brave iOS keeps a web <video> playing when the browser leaves the foreground (Settings → Media → "Enable
+// Background Audio") by re-playing a video the OS paused. But it only hooks the <video> elements it SEES:
+// those present at document start, or a MutationObserver addedNode that IS a <video> (brave-core
+// ios/browser/web/media/resources/media_backgrounding.ts). React inserts a player's <video> inside its
+// freshly mounted wrapper in one go, so the video itself is never an added node and Brave never hooks it —
+// measured on iPad Brave 2026-09-14: the shared video stopped the moment the app was left; inserted as a
+// node of its own it kept playing, locked screen included.
+//
+// Re-inserting the element at its own position makes it an added node without changing the DOM React
+// manages. Playback is not interrupted: a media element removed and re-inserted within the same task is
+// back in the document by the next stable state, so the spec's pause-on-removal never runs.
+//
+// Only a <video> needs this: an <audio> already keeps playing in the background on its own.
+
+const exposed = new WeakSet<HTMLVideoElement>();
+
+/** Make a mounted <video> visible to the browser's background-play hook. Idempotent per element, so it is
+ *  safe from a ref callback that React calls on every render. */
+export function exposeVideoToBackgroundPlay(video: HTMLVideoElement | null): void {
+  if (!video || exposed.has(video) || !video.parentNode) return;
+  exposed.add(video);
+  video.parentNode.insertBefore(video, video.nextSibling);
+}