Overview so far of our trip - comments welcome!

There are so many hill towns to explore. I liked Montepulciano a lot and would recommend that. Pisa and Lucca could be done in a day, so I would add other towns to the 3rd-7th portion of the trip, but that's me, I like to visit more places and spend less time, especially since you have a car for that part of the trip. Have a wonderful trip.

San Gimignano is an easy reach from Siena but if you are headed north you could stop by there on the way. Also Voletrra is close to San Gimignano.

Vinci is also near San Gimi.

North of Lucca up the Serchio River, nice drive and beautiful country.

On the coast Portovenere.

How about north of Lucca? We were thinking of driving up to Vernazza and CT - but it is 2-3 hours from Pisa. What is it like east of CT countryside, and north of Lucca? I hardly ever see anything written about that area (or at least nothing has popped out at me).

I don't know why you said it would have been better to fly to Venice. If anything you could have flown to Florence and out of Rome (or viceversa). I don't see Venice at all in your plans.

Given your current arrangements I see you have 8 nights (5/31 - 6/7) before you go to Florence, then Florence and Rome.
The first 8 nights from flight arrival to Florence arrival:
3 nights in Maremma (Pitigliano is ok): use as base to visit the Maremma, including Argentario peninsula
4 nights in Siena province: base for a lot of small towns in Tuscany
1 night add to Florence, just go one day earlier (same apartment if you can). Visit Lucca & Pisa from Florence by train using the extra day gained.

The above minimize lodging changes. You have a car, so you can use as a base (a hub) and drive in all directions around from your hub.

If you are interested in CT, then go! Can you imagine going home and saying you didn't do it because it was 2 hours away?

5/31 - Arrive Rome 10:30 am - Pick up Car Rental (still have to book this) - First night (booked) in Pitigliano
Can you extend here, and add a second night? Visit Orvieto perhaps on the 1st?
6/2 - Drive to Montepulciano for 2 nights
6/4 - Drive to Siena for 2 nights
6/6 - Drive to Lucca for 2 nights
6/8 - Drive to Florence (2 nights) and drop off car - apartment (reserved) in Oltarno neighborhood
6/10 - Train to Rome (4 nights) - AirBnB (reserved) in Monti neighborhood
I would book an evening train to give you the full day to see more of Florence.
6/14 - Fly out of Rome

I am surprised to see you travel in 10 days & are yet to secure a rental car - be prepared to pay a premium rate.

I would have given myself another day more in Rome at the end but does not sound like that can be reworked

Honestly for those open days you could easily keep your outside of Siena base and day trip by car to different hill towns each day. Siena is excellently positioned for doing so and would be nice to have a home base rather than moving about so much
If you did want to add the cinque Terre for 2 or 3 nights you could definitely do so and in that case would stay there

According to viamichelin.com, driving from FCO to Pitigliano will take at least 2.5 hours. Driving jetlagged/sleep-deprived is equivalent to driving drunk. It's a bad idea, even when you know the car and the roads.

Why not spend your 3 days as pedestrians in Rome when you arrive and while you get over the time change. Then do your driving, return the car in Florence. If you have an early flight out of Rome on the 14th and need to spend the night there, you could start in Florence, then rent a car for the rest and return it in Rome.

Thank you everyone for your input. The open ended (not itinerary and booked ahead) is that part, I think, that will be the most fun. We did end up securing 2 nights in Siena just outside the walls - I'm ok with that because I just did not want to deal with parking. The place is run by a young couple, we have the entire flat to ourselves, they can give us personal advice on sites, etc - so despite not being 'inside the walls' we will be able to enjoy late evenings there and catch a cab home, and spend the mornings there too as it it is close by.