Tuscany with young children

Ciao!
Many thanks for contacting me,
Happy to hear you like my blog
All the Tuscan towns and villages you mention are wonderful, great you’re including them in your itinerary! Other must see villages I would visit are San Gimignano, Montalcino, Pienza and Montepulciano.
San Gimignano is the town by medieval towers, a real gem! Montalcino, Pienza, and Montepulciano are sensational hill towns south of Siena, in one of the most scenic areas of Tuscany, the Val d’Orcia. Also take a look at my list of the top villages to visit in Tuscany.
Considering the destinations you mention and the ones I suggest, I think a good location as a base would be Cortona, in Val d’Orcia, or in Siena itself.
It’s not so easy to find a holiday rental with a small garden in a town’s center. A place where I often stay is Bagno Vignoni, small stupendous village near Pienza. Here is a picturesque square with an ancient thermal pool at its center. Hotel Le Terme faces the square and also offers accommodations in private apartments with garden at one side behind the piazza. Take a look here. The place is full of peace. 
For a more lively piazza tough I’d suggest the beautiful Cortona. This house with private garden in the heart of Cortona can be a possibility, together with this other lovely private house with little garden still in Cortona. 
Here is a useful post to using Cortona as your home base in Tuscany with a list of the best day trips you could enjoy.
Do contact me again if you wish more suggestions,
Enjoy your vacation!
Elena from Florence