refactor(frontend): each page fills the floating header itself; the header knows no page

The header used to pick its pills by `page === …` and import five module controls, and Plex's
Discover/Browse toggle needed a context bridge to reach the sub-view that lives in PlexBrowse.
Now the header is an empty band: PageShell hands its root box out as a slot, a page portals its
own pills into it (PageHeader), and the palette's Jump-to pill shows where no page does
(PageHeaderDefault, rendered inside the page's Suspense so a loading page shows no stand-in).
Plex's toggles take the sub-view as a prop; PlexViewBridge and its context are gone, and with
them the header's five layer suppressions.

No visible change: every header pill's box measured identical against the previous build on eight
pages, in Hungarian and English.

frontend/src/modules/channels/Channels.tsx

@@ -64,6 +64,7 @@ import { useFeedFilters, useFeedFiltersActions } from "../feed";
 import { useWizard } from "../../features/wizard/wizardContext";
 import { useMe } from "../../lib/useMe";
 import { useTags } from "../../lib/useTags";
+import ChannelsHeader from "./ChannelsHeader";
 
 export type ChannelsView = "subscribed" | "discovery" | "hidden" | "blocked";
 
@@ -687,6 +688,7 @@ export default function Channels() {
 
   return (
     <>
+      <ChannelsHeader />
       {pageMenuItems.length > 0 && (
         <PageMenu ariaLabel={t("common.pageMenu")} items={pageMenuItems} />
       )}

frontend/src/modules/channels/ChannelsHeader.tsx

@@ -0,0 +1,26 @@
+import { useTranslation } from "react-i18next";
+import { useMe } from "../../lib/useMe";
+import { PageHeader } from "../../components/ui/PageShell";
+import SearchBar from "../../features/header/SearchBar";
+import { blurActive } from "../../features/header/blurActive";
+import { useChannels, useChannelsActions } from "./channelsContext";
+import SyncStatus from "./SyncStatus";
+
+// The channel manager's pills in the floating header: the name filter and the per-user sync chip.
+export default function ChannelsHeader() {
+  const { t } = useTranslation();
+  const me = useMe();
+  const { search } = useChannels();
+  const { setSearch, goToFullHistory } = useChannelsActions();
+  return (
+    <PageHeader>
+      <SearchBar
+        value={search}
+        placeholder={t("channels.searchPlaceholder")}
+        onChange={setSearch}
+        onGo={blurActive}
+      />
+      <SyncStatus isAdmin={me.role === "admin"} onGoToFullHistory={goToFullHistory} />
+    </PageHeader>
+  );
+}

frontend/src/modules/channels/index.ts

@@ -1,3 +1,5 @@
-// The channel manager's public face: its store, mounted at the root and read app-wide.
+// The channel manager's public face: its store, mounted at the root and read app-wide, and the sync
+// chip, which the feed's header shows too.
 export { ChannelsProvider } from "./ChannelsProvider";
 export { useChannels, useChannelsActions } from "./channelsContext";
+export { default as SyncStatus } from "./SyncStatus";