I think the question is how much exploration vs. how much relaxing vacation are you looking for?

Are you looking for relaxing beach days and, if so, how much of the vacation do you want to commit to that? Positano would fit that bill with the warning that Positano (specifically) and Amalfi Coast (generally) are quite crowded in the summer months. The beaches are small & crowded and you'll want to reserve at a beach club for spots as the "free" beach is next to non-existent (there is a specially marked off area for locals which you won't qualify for). There are some good exploration from Positano as a base (Capri, Ischia, Pompeii, up-and-down the Amalfi Coast) but movement is very slow by car, I'd recommend those be explored by boat/ferry - Positano has a number of routes that go to and from Naples, the Amalfi Coast and Capri).

So, the split you propose, 6 Positano/4 Naples would give you relax time in Positano, some exploration (Capri is quite interesting, Pompeii is a great way to experience ancient Roman history) and then some city exploration in Naples.

Alternatively, if one was flying into Rome and wanted what you initially proposed, Rome, Florence and the beaches around Viareggio (near Florence/Pisa) would fit the bill. The stretch of beach from Lerici down to Viareggio is a huge sandy beach with lots of alternatives in terms of how developed vs. wild.

I posted this video from Nicki Positano (a youtuber who lives in Positano with her family) in an earlier post. At around 20 minutes, she and her friend (who runs a ferry business) talk about how crowded Amalfi Coast/Positano is with tourists this last summer (certainly up much from pre-pandemic levels as a lot of people were checking off their bucket list). You might want to view it for some idea of what it is like in June. Personally, I like Amalfi in late September/October. Temps good but much less crowded. But understandable with kids that the summer is what's available (my wife's cousin and his family are coming to visit us in Italy in July/August for exactly those reasons).