This is an interesting comment since part of the posters criteria was relaxing (nice beach, quiet location, excellent service, good status recognition). and the suggestion of Coconut Point would meet this criteria very well.

And the only property that was suggested that meets these criteria on Happys list would be the Timeshares in Key West. And while these do have the Hyatt name on them, they will not earn you any GP points, stay credits, amenities, or other status recognition.

There is not a single perfect Hyatt property in Florida that will meet all the criteria. If your main focus is to relax, then look at the Resorts at Coconut Point or Key West. Otherwise you should determine which area of Florida you are most interested in.