Ciao!
Will you rent a car to move around? Let's assume you do have one which will make moving around easier and faster.
Even with a car, you're trying to do too much - you won't be visit every place on your list. This is what I'd recommend for the part in Tuscany ( 2 full days in Rome are fine).
6/21 - Florence
6/22 - Florence
6/23 - Siena - drive down to Siena through Chianti so you can DRIVE through Chianti. Don't make more than 1 stop, if you want to arrive and see Siena....
6/24 - PISA instead of Lucca - visit Monteriggioni and San Gimignano along the way to Pisa. Spend the rest of the day in Pisa.
6/25 - LUCCA/PIENZA - you could head north to Lucca and spend the morning there.... and then drive back down to southern Tuscany. That way you can stop in Montalcino in the afternoon, ending up in Pienza for the night.
6/26 - visit MONTEPULCIANO in the morning, then head down to Rome.
I've managed to fit in everything except for Vinci, CT and Cortona but they are the ones most out of the way as you move around... if you had at least 2 more days, you could fit in CT. Distances in Italy seem "short" but roads are 2 lanes and curvy over hills, so you don't drive fast on them. Each place has a lot of things to see... so this is really already a full itinerary, you actually could use a whole day just for Chianti and another full day for Siena, so trying to do both on the same day is already a lot!
In Italy, you will make the most of everything you see if you take your time to see each place and enjoy it - eat gelato, eat full meals and "trattoria" and restaurants which are part of the experience as well!
If you weren't thinking of driving.... then lots of places are off the itinerary, including Chianti, Montalcino, Pienza.... so do tell me you're thinking of driving!

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