I have been to Grand Cayman several times including both cruises and land vacations. I have never rented a car. Taxis are plentiful and can fit larger families. They can be taken to or from any hotel on Seven Mile Beach. We personally walked around and even from the port to the beach although it is a hike.
There is another trick I have done on other islands. I really liked hiring a taxi for an island tour. They take you around and show you the tourist spots as well as others and let you stop for as long as you want. I haven't done it on Grand Caymen, but I am sure it could be done.
Have fun.
DG