Your plan is not bad as a sketch, but many of the details embody mistaken assumptions about the infrastructure. I can't get all of them in one reply. This is a very crowded area in a very crowded time of year. You are going to have to take your sixth choice of hotel, except maybe in Salerno, a pefectly pleasant modern small city with less tourism but less to see. (Paestum is most accessible from Salerno, but it gets much less tourism. We'll see it on a future trip, because we slept in Sorrento.)
You are right to prioritize Pompeii, and many posters here do it the way you propose. However, it is baking hot and sunny at midday, and you might wish (since you're sleeping in the area) to get there near opening time as a daytrip.
(I) have plenty of money, and agree that having four guests makes the private transportation you propose several times on and off the water a reasonable splurge. But this kind of transport requires advance booking or a luxury hotel with a concierge. It may be hard to fit all your luggage and four guests in a large sedan like mist bookings provide. I mean typically a Mercedes sedan.
The Circumvesuviana is a griity, crowded surface transit line that runs from Naples to Sorrento, with stops that include several ancient excavations nearby. The long-distance train company serves Naples and Salerno, with a different, intermediate stop, in Modern Pompeii, some 1/2 mile from the ancient excavation you want to see.
You don't usually take the ferry to Sorrento and the another ferry to Capri. That adds up to too much time and ticket buying. In season there are ferries from Salerno to Capri. I prefer Sorrento [Edit-Corrected city name] to sleep in because of good travel connections, but you may be too late to get rooms in Sorrento. We took a luxury hotel at the top of our budget because we booked Sorrento only five months ahead. I'm surprised you haven't mentioned Positano or Ravello, two lovely sights.
Edit: I'm not questioning your personal preferences, but tell us why are you staying in Grottaferrata. (I've never been there.) All I mean is, is it to visit Tivoli while you have the car? I'm questioning suffering Rome traffic just to get to Naples in a car. It's not desirable to arrive in Naples in a car because the city is so awful for auto traffic, and the car is useless there. (I think you are not sleeping in Naples. We spent one night there, and it was barely enough in a city rich in history and civilization. Not to mention, pizza.) More to the point, you can take an effortless, advance-purchase Frecciarossa premium-service rail ticket ALL the way from Florence to Salerno. Unfortunately, you are too late maybe for the dynamically-priced lowest fares. (Even though Salerno is not MY choice for an Amalfi Coast area stay. As I said, I"m trying to respect YOUR city choices.) I have driven from Tuscany to FCO, and it's just not a special superhighway drive.
Note that there are some newsboard posters who LOVE Salerno. Other newsboard posters LOVE Sorrento. You are quite entitled to favor posts that support your own interests! But please understand that staying in Salerno is directly related to issues like visiting Paestum and choosing one particular access to Pompeii Scavi (ancient excavation.) It is, for example, much farther from Capri.
You might benefit from a look at this area on Google Maps, and then an online PDF of the Circumvesuviana and Trenitalia lines in the area. It makes the infrastructure easier to understand.