You could add WordPress default search block to the header template.
But if you need an option like you mentioned, there is no option like that for the moment. We do plan to add our own search bar in our Gutenverse plugin, and we will take note on this feature.
If you still have any other question, please let me know.
I installed the Ivory Search plugin and configured as per the author’s directions which basically turns the search icon on – on the primary menu they are saying if it isn’t showing it is a theme issue. Can you look at my site and let me know what is being blocked/how to fix?
How do you put the search on the menu? Do you use a shortcode block? I try to replicate and use a shortcode block to put the shortcode and the search appeared