The most child friendly place I have found in Europe as far as beach resort is the Costa Navarino in Greece. It is affiliated to Starwood and it has two big hotels in one beach front complex (the westin and the romanos). It is not Aman, but it is certainly a nice place where to stay and it is for sure a place that your kids will love as no other beach resort.
Food varies from good to very good...they have 15 restaurants. my fav are Inbi (japanese) and flame (grill).

Forget about Italy, I live here and I keep going abroad every summer given the complete lack of infrastructures for families.

Now that the pool is complete Sveti Stefan is for sure a nice one to try and I am probably give it a shot next summer. I have already inquired and the best option might be a small dependance close to Villa Milocer made of two suite...very close to the beach. You could take the whole place for you....but again, it is not Italy.

Someone like the Verdura resort in Sicily by Roccoforte, give it a look.
Sardinia is a nice place, lovely beach...but still I have not found the right resort...give a look to Forte Village just in case.