feat(nav): the live YouTube search and Home's budget are named in the address

A YouTube search is `?yt=<term>` (+ `ytn` for a non-default count), so a reload, a link
or Back/Forward reopens it. It runs by itself only where it costs no quota: the free page
scrape, a search asked for in this document, or results still cached. An API search
reopened any other way shows its term and waits for "Search YouTube again".

Home's time budget is `hb`; the question's draw stays on the history entry.

An e2e spec opens every page's bookmark in a fresh browser context; CLAUDE.md and
docs/development.md name the module URL codec as the place for view state.

frontend/src/modules/scheduler/Scheduler.dom.test.tsx

@@ -380,6 +380,8 @@ describe("a row's history", () => {
           closeChannel: () => {},
           enterYtSearch: () => {},
           exitYtSearch: () => {},
+          askYtSearch: () => {},
+          setYtCount: () => {},
         }}
       >
         <QueryClientProvider client={qc}>