welcome to CC.
Everything you said about Labadee applies to all private islands - with the exception that Labadee is not an island but an isolated beach. Be assured that if there were sights "outside" Royal Caribbean's property, excursions would be offered. The sad fact is that Haiti is the poorest country in the Americas and has a long history of violence and civil unrest.
Of course Labadee and Grand Cayman cannot be compared. But then the cruise line had changed the itinerary for a very good reason. No cruise ships are allowed to visit Grand Cayman at this time, and you can be sure that as soon as local authorities allow cruise ships to return, the cruise lines will return.