Thanks for this - turning off the search bar from the layout button does indeed take the unwanted buttons with it, but there is then no way to carry out a search.
I was hoping there would be a 'search' action that I could then add to a toolbar, but if there is I can't find it (?)
It appears that those buttons are 'hard-wired' into the search bar and you can't search unless they also come along for the ride.
This is with the latest version, Calibre 8.16.1, and no plugins.