Oh, didn't realize you have been to Florence and maybe some of these other places before.
That makes a big difference since you know the lay of the land and may not need to hit some of the "most sees" if for this trip you are primarily thinking of what the kids will enjoy.
Yes, if you leave Venice off and have 4 or possibly nights without really adding more nights, I would recommend renting a car and staying in one of the smallish towns like: Volterra, Montalcino, Pienza, Montepulciano ; they would be ideal with a car.
We stayed in Pienza and loved it but it is very small ; with teens and more things to do Montepulciano or Volterra would probably be the 2 I would look at the most ; these towns have bus service but is designed with commuters not tourists in mind so the schedules are not going to be as frequent as you would like ; that is why the car is such a benefit.
If you prefer a more remote Agristristmo the benefits are obvious, the downside is nothing is within walking distance. You know your family better than us to determine if that will work out well or not compared to a town like Siena where you can easily walk about. The same towns I mentioned above are all smaller than Siena and all have Agristristmo's around them.
You could arrive in Florence on the train. Maybe store your bags, see some sights in town then rent the car from Florence center and return the rental car to the airport when flying out.
Of course if you can add a couple of nights ; why not keep Venice in the plan ; I am sure the teens would love to see Venice.
Budget may say otherwise though especially with the budget busting costs of staying in the Amalfi Coast in August.
If you do decide to keep Venice I would drive from whatever town you choose in Tuscany to Venice ; dropping the rental off when arriving rather than trying to navigate back to Florence with the car in order to take a train. Would be a 4 hour drive roughly.