Comment to Messenger Icon
I did some customizations to the menus, sorry. Redo the changes in toolbar and check them in "Toolba Member". That's where messenger is located by default. I moved it to another menu. In your case messenger should be active. Check settings by clicking the pen.
If you want to turn search completely off, just deactivate it in Toolbar Site menu.
T
I did that in ...path-to-site/studio/builder_menu.php?page=items&module=system&set=sys_toolbar_site but search icon still present....
Try to clear cache.