My family (including 3 kids of varying school ages) and I had a wonderful vacation in Positano. Beautiful, self-contained village cascading down to the sea, friendly people, lots to explore nearby by sea and by land, no cars in the village, and so on. If you've been there, and you know what I mean, could you suggest something comparable for a late summer holiday?
Last summer we spent a week in Menaggio on Lake Como. We were 2 small kids, 2 parents, an Aunt and an elderly grandmother. The town of Menaggio was a delight with lots of narrow streets and alleys to explore. Swimming in the lake was fun. But what really made it was that it was a short walk to the ferry dock, and from there it was easy to ride to other towns on the lake. If someone wanted to just hang out in Menaggio they could, or if they wanted to escape on a ferry across the lake they could do that too. I went into it a little afraid of the "big family vacation", but there were enough options to keep everyone happy and opportunities for individuals to "escape" on their own for several hours. In short, we had a great vacation and I would not hesitate to recommend it. Fond memories.
Are you staying with Italy or open to another country. Yes, Positano is fantastic in my opinion, we were there last June for a week.