This could give an idea of what people is looking, and how to focus our mapping efforts.
For example, a country community could be focused in mapping highways, but its users (not mappers) are looking for political division (neighbourhoods, cities, places).
A list like that could give an idea about what to map next. Currently, it is based on our assumptions as mappers, but not as users.
Way back we included language in the Nominatim ToS to allow use of the query strings for such purposes (at the time I was thinking about missing data), that then back in 2018 made itself in the the OSMFs provacy policy Privacy Policy - OpenStreetMap Foundation This isn’t the 1st request for access to the data (historically there has been at least a handful of researchers that wanted access).
However nobody has ever done the work to produce an anonymized data set from the input strings that could be made public. This isn’t a trivial task, and it isn’t clear if it can even be done in a reasonable fashion.
Not that I know of (obvious hen and egg issue that you could only have that if you had a method to anonymize the inputs).
The way to move forward would be for a interested data analyst to sign a NDA with the OSMF after agreement has been reached with the relevant working groups (LWG, EWG and OWG) and then get access to the raw data for developing a solution.