That's not really what's going on though.
The same behaviour happens when you turn off "Location History", Google Maps stops showing the last few searches in its search box. That's literally a matter of local storage that's available in any decent GUI library.
It's really stupid but it's the only reason why I keep Location History turned on. I don't really want Google to keep my Location History, but when I'm on the road, it's super inconvenient to have to type full address over and over again.
I wonder if with the reverse engineering LLMs allow us, I could patch Google Maps to behave like a normal program with a normal search input field.