feat(plex): subtitles rise above the control bar while it shows

On an iPad the bar covered the text whenever it was up: the browser places a
native cue by its line, which nothing animates. The track now stays loaded as
hidden and PlexSubtitles draws the active cues, measured against the bar and
lifted above it in 200 ms, back down when it hides. The viewer's size, colour,
background and outline settings style the text directly. Also fixes a test
typing slip from the search pill change.

docs/DESIGN.md

@@ -713,3 +713,4 @@ of a full-width panel and nothing on the right.
 | 2026-10-06 | `player` focus | A dialog that is a player opens focused on its panel (the YouTube player's rule, now `Modal`'s too); on the media element itself, Space and the arrows stay the element's own | `close`, or the media element | On the ✕ Space closed the player; on the element the browser's own controls acted on the same key as the player's (one Space played and paused, measured in Chrome) |
 | 2026-10-08 | Captions on touch | A `Tooltip` opens for a mouse and a key's focus only, never for a finger | A caption on tap; a long-press caption | On an iPad the tap that showed the caption was taken as a hover and lost its click (every rail button took two taps), and no finger sends the mouseleave that hides it; the rail's labels setting names the buttons on touch |
 | 2026-10-08 | Header search on touch | No key cap on a touch screen; the pill grows to 52 px so a ring shows round the 44 px Go; Go drops to its icon by the column's width | Keep the key cap; a 44 px Go overhanging a 40 px pill | On a tablet the 44 px key cap and Go sat on the pill's edge and cut the placeholder; an overhanging Go breaks the nested radii |
+| 2026-10-08 | Plex subtitles and the control bar | The player draws the cues itself; while the bar shows they rise above it (200 ms), then settle back | The browser's own cues, moved by `line` | On an iPad the bar covered the text; a native cue cannot be animated or measured, and our layer behaves the same on every engine |