Thomas,
I am pretty certain the mostly do. And, the tracking map would make it pretty clear if a cart was short cutting to the rest room or something similar.
Agree short grass and non-water hazards reduce lost balls and lost ball searching (well, water reduces search time in most cases, and some speed of play experts recommend lots of water hazards, at least compared to long, native grasses for that reason)
The thing golf faces is the reduction in water use, which leads to converting more formerly turfed areas to long, native-ish areas. At La Costa, for instance, we found an unirrigated area of paspalum that could have been left, and watered only once per week or less to keep. However, the state of California was paying $XX per acre for NO irrigation, so we had to replant with another species.
Similarly, I find here in the south, that common Bermuda (sometimes it seems the oldest, worst turf variety you can find, like BlackJack Bermuda) establishes quickly and works good enough as unmown grass at 3-12", again perhaps watered a few times a year.
In typical golf fashion, the idea of creating a shorter, no water turf will probably be handled using more complexity, genetic research, etc. Otherwise, we are stuck with the current trends to turf reduction.