Taking public transportation in Brooklyn/Manhattan at night is not safe for a girl. Taking public is also not time/cost efficient. Therefore it makes more sense to take a car. I’ve never felt that driving on a date is not tznius. Other aspects I have felt are not so tznius such as the girls getting decked out(over the top) but the actual driving was never an issue.
And as LU pointed out, the girls parents usually want to meet the guy first so it would be weird to then travel on a bus/subway together. It’s also quite untznius taking public transportation in general during the summer months.
On the topic of activities, I’ve found that they really break the ice and the boy/girl give the most accurate depiction of themselves during those times. It also allows you to see many aspects of the person which you would otherwise have a hard time finding out. In lobbies and similar venues there is too much pressure to be a stiff and proper.