Getting rid of "Advanced Search" in a Grid - SC8

So I have a grid app (configured to start in grid mode) with the ‘Search’ not displayed in neither top nor bottom toolbars from Grid => Toobar. This is because I only want to do quick search on this grid which is already displayed.
The Search, when displayed, shows up as “Advanced Search” to the end user.

For all I know, I simply copied the desktop toolbar config to mobile and all of a sudden the Advanced Search shows up in the running grid, despite it configured hidden in Grid => Toolbar. Worse, it shows up in both top and bottom toolbars on both desktop and mobile.

Asides simply removing search entirely from Grid => Grid Modules, is there any other place to configure this?