We now have 3 search UX variations:
* Header search
* Welcome banner search
* Icon search
And within each of these there is a `<SearchMenu />`
component with an input that needs to be focused based
on either Ctrl+F or `/` shortcuts. This commit makes
sure that each has a unique ID, and moves the functionality
of determining the "current" input ID and focusing the
input to the search service.
This fixes issues like where pressing Ctrl+F twice on
the header search would not reveal the regular browser
search.
