For some reason, this feature (Kagi Settings > Search > General > Search Options Always Open) does not work on mobile browsers. If I select "request desktop site" it will honor this setting, but when I return to the mobile site it will not.
It is VERY handy to have these options always available. Yes, it takes up a bit more room, but it should be a choice for the user.
In its simplest form, simply honor this already existing setting on mobile devices. But if a distinction is needed, you could add a "desktop" and "mobile" checkbox sub-option.