Best Caribbean islands for families

Hi, we recently spent a week in Grand Cayman with our two young kids (preschool and elementary school age) and loved it! I now want to explore more of the Caribbean. Which island should we go to next?
(Some on my interest list include Aruba, Anguilla, St. John USVI)

Of note, we are flying from Miami, and direct commercial flights are preferred (no small seaplanes).

Priorities include: safe (low crime), low disease risk (GI and mosquito), nice beaches, but more to do than beaches (eg. Snorkeling, good food, history, culture). Also, budget-friendly options available are a plus. We made grand cayman work on a budget (stayed at the locale, rented a car, mostly ate at local dives).

We are not the type to go to an all-inclusive and never leave the resort; but I’d be open to it for like 3 nights max; eg Nickelodeon resort in punta Cana with the kids seems cute but expensive.

I have otherwise not explored much of the Caribbean; I have previously been to PR as a kid, Ocho Ríos, Jamaica on a cruise (brief), Playa del Carmen, Mexico on a cruise (20yr ago with my parents).

We love changing it up and going to new places. Any suggestions are welcome. Thank you in advance!