I did the med way back in 2007 (inaugural year, so they were still figuring things out). First, you have been to Europe so know that the streets/sidewalks are not smooth and many strollers, well, are no good at navigating them. The days are long. And I mean long. Off the ship early, back late. Lots of walking and in one port lots and lots of stairs. Anyone else get to do the stairs from the Sorrento to the ferry to Capri? That was fun watching the woman with the 4 inch heels navigate down. Rome was a full day and most of the kids ages 3-7 were asleep before the end of it with parents looking pretty miserable having to carry/miss out on things because the kids were not behaving after being our of their element for 6+ hours. At the time, and please, many of these excursions have changed and/or been tweaked, it did not seem like many of the tours were kid friendly. We went to the crypt in Palermo, Sicily with half a bus of kids. Heck, I remember asking a CM if they could make an adult only bus a few times and now
DCL offers adult only excursions, no we did not mind the kids, but the parents of the kids sometimes lacked respect for the rest of our time.
Here is what I say, you know your kids. Sounds like y'all are pretty well traveled and that they are used to new/different/unfamiliar surroundings. I will also say that there is very little that your kids will remember from Europe at this age. My parents swear they took me places before I was 6 that I have very little memory of. I barely remember going to Disney World when I was 7.
The Caribbean beaches are beautiful, as is the water. However, some of the cities are dirtier than others. It is part of the economy of where you are visiting. I guess I would need to know what your BFF considers "dirty." Dirty, as in Paris was dirtier than London, imho. The Southern Caribbean is probably cleaner than the rest. Cozumel is not the cleanest town. But the beaches are pretty. And they have lots of places to get great margaritas.

Grand Cayman seemed very very clean. Port days are shorter. More activities for children. But again, you know your kids and the excursions in the Med may have changed tremendously from when we went. Yes, we still want to go again, it was an amazing trip! But, I have other places to visit first now!
I would suggest private excursions for everywhere except possibly Rome, but I only say that because of the penchant to strike there, lol. Not sure if it has been as much of an issue lately. When we went, there had been strikes the previous months and more were being threatened, which would have meant roadways would come to a standstill. This way, you can control not only what you and your family does, but the timing. If you know your son is only good for 4 hours, then you can probably make it work this way. And as others have stated, you could leave the kids on the ship at some of the ports. Also, private excursions may get you off the ship and on your way faster. Sometimes, it felt like we waited so long to get to the bus to leave on the DCL excursions. And they were pricey. Worth it mostly, but pricey.