Month and year of travel?
georgyyg, this is partly an opinion, but what they are trying to tell you is that the Amalfi Coast has, geographically, nothing to do at all with Rome or Sicily. It is a troublesome distraction, because it is relatively hard to get to, and hard to get elsewhere - because it's always crowded, and it's off the regular transportation routes. It is NOT undeveloped or undiscovered, I'm not saying that. The United States has dozens and dozens of beaches that are superior to anything on the Amalfi Coast. The TOWNS are attractive, I'm simply talking about the beach property. I was looking forward to seeing Positano, but I found it a shadow of a luxury for 1950's wealthy folks, with the same high-end fashion and cosmetic boutiques you can visit in Paris, London, or Rome. There are lovely cliff vistas of the ocean, and Ravello still has some charm. But don't try to cram this in-It doesn't pay off for the time spent getting there on your schedule.
Your kids will have another chance to visit the area (especially Pompeii), if they are interested, in the future. If you allocate 5 nights to Sicily, you should spend the rest of the time in Rome. Don't forget to subtract travel days.