refactor(plex): split browse.py god-module into per-concern routers (E3 S1)

routes/plex/browse.py was a 1693-line god-module (libraries + facets +
collections + playlists + show/item detail + people). Split it into focused
sibling routers under the same /api/plex prefix — pure code motion, no SQL /
route-path / response-shape changes:

  browse.py  1693 -> 469  (list_libraries, unified_library, facets)
  collections.py  (new 253)  browse + admin collection editor; owns _collection_source
  playlists.py    (new 405)  the user's orderable playlists
  detail.py       (new 438)  show/item detail + enrichment; _lib_key lives here
  people.py       (new 139)  cast/crew chips + person filmography

Cross-module helpers _norm_lang + _content_libs move to _shared.py (their one
home); detail imports _collection_source from collections (one-way, no cycle).
__init__.py includes the four new routers. Three DB-lane tests follow the moved
symbols (import path for _content_libs; monkeypatch targets for PlexClient /
_show_enrichment now on detail).

Behaviour-equivalent: ruff clean, 348 DB-lane pass (unchanged), parity net green.

backend/app/routes/plex/_shared.py

@@ -3,10 +3,12 @@
 import logging
 
 from fastapi import HTTPException
+from sqlalchemy import select
 from sqlalchemy.orm import Session
 
 from app.models import (
     PlexItem,
+    PlexLibrary,
 )
 from app.plex.lookups import by_rating_key
 
@@ -18,3 +20,22 @@ def item_or_404(db: Session, rating_key: str) -> PlexItem:
     if it is None:
         raise HTTPException(status_code=404, detail="Unknown Plex item")
     return it
+
+
+def _norm_lang(lang: str | None) -> str:
+    """The overlay has an EN base + an optional HU twin; anything other than `hu` serves EN."""
+    return "hu" if lang == "hu" else "en"
+
+
+def _content_libs(db: Session) -> tuple[list[int], list[int]]:
+    """The BROWSABLE content libraries, partitioned into (movie_lib_ids, show_lib_ids) — the single
+    source of the browse scope. Excludes HU overlay libraries (`overlay_of` set): those are never
+    browsed on their own, they only lend their Hungarian metadata to the EN twin's rows (E5 S5)."""
+    libs = db.execute(
+        select(PlexLibrary).where(
+            PlexLibrary.enabled.is_(True), PlexLibrary.overlay_of.is_(None)
+        )
+    ).scalars().all()
+    movie_lib_ids = [lb.id for lb in libs if lb.kind == "movie"]
+    show_lib_ids = [lb.id for lb in libs if lb.kind == "show"]
+    return movie_lib_ids, show_lib_ids