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.

As in, Google's AI plans explicitly specify the amount of storage you're buying.

No, that doesn't apply to free usage - but that old adage about "If the product is free, are you sure you're not the product?" applies.