Coral Beach is your only option for a turnkey beach club. It's not a great beach - has had some erosion due to being on the southern end of Seven Mile Beach (SMB).
All beaches in the Cayman Islands are public, but there is only one Public Beach, which is a stretch of Seven Mile Beach, pretty much in the middle of it, north/south. There are chair and umbrella rentals there, plus lots of watersport options. Restrooms there are very clean, and you can change in them, based on my experiences.
There are plenty of food options there - food trucks and locals set up their stalls, especially when cruise visitors are in. The quality of the beach there is far better than Coral Beach, too. of you want something nicer, you can walk up the beach a stretch to the Kimpton where there is a beach bar/restaurant called Coccoloba - https://www.coccolobacaymanislands.com/
If you just want to pick up some food and drink at the supermarket and head to a beach, there are two more sections of Seven Mile Beach that are worth visiting - Cemetery Beach and Governor's Beach - there are no facilities at either of them, but there are often porta-potties at Governor's - those are far less crowded, of course.