We've taken our kids to Italy all their lives starting as early as 1 year old. Anyway, the second trip we took to Italy with our oldest was to the Maremma area of Tuscany. A little wilder, a little less overrun but an easy drive from FCO. We used https://www.parrina.it/en/ as our base and then made regular trips to small towns like Orbetello, Porto Santo Stefano, Porto Ercole (swankier), little hill towns like Magliano in Toscana, etc. Lots of great wineries in the area and they're totally cool with you bringing by kids... just keep them corraled. Beach wise, there are beach clubs just west of Ansedonia... dark/volcanic sand beaches so a little darker sand but its a cool area... you can even stay in the house where Turandot was written (you know the song, trust me), see ancient ruins by the beach, etc.
Pictures from 2010 (!!!)
Pool at La Parrina
La Parrina (shop at agriturismo)
Small beach below a restaurant near Porto Santo Stefano
Two views of the beach at Ansedonia from above and on the beach
Beach club in Talamone