I dont really understand the two first suggestions ?
The third one IMHO does not compromise functionality. A italic dimmed out placeholder "Search through installed applications, settings, and actions" makes it clear what the search you type in is for, and that it is out of the way. You can ignore it but its there.
It also seem very simple to do.

@teamcons the first point is more of a quirk of how GTK3 works. In gtk3 when a text entry gains focus its placeholder is hidden. In gtk4 the placeholder is hidden after you start typing

The second point describes how you can reveal this information since the search bar is on the same page as the app grid. Alternatively the search page could be an entirely different page hidden behind an icon for example, but I think being able to start typing right away is a pretty essential feature. So we can put keyboard focus on the first app in the grid and then if you manually move focus to the search entry yourself, replace the grid with this information.

However there is a question of how to hide this and get back to the app grid so I dunno. I think that's maybe somewhere I stalled out in the prototype I did