Thank you for the question.
We don’t have such a feature at all:
“It’s meant as a shortcut to quickly load the map for a specific location without using Advanced Search. ” I need more details to advise if this can be done as custom development. Maybe you can email me an example via email (i got your email too but decided to reply here)
And, to confirm – you need the WordPress search widget show on mobile? Or you wish something else?
On Mobile Advanced Search is different than on web – so even if you have advanced Search type 1 on web, on mobile it will be replaced by the Mobile Advanced Search in mobile view.