I thought that staying in/near Siena, using that as our base, and driving to any towns nearby would already give the country experience.
Some historical perspective on Florence and Siena might clear up any confusion. Florence and Siena are serious cities. Montepulciano and Montalcino, less serious.
Even though they are in Tuscany, I don't view Florence or Siena as "country" experiences. Yes, if situated properly, you may be able to sneak a view of the country from some isolated, elevated perch. Both are smallish cities, for sure, when compared to Rome and Milan, but they are dense, busy cities with some serious traffic issues.
Unlike most of Florence, Siena is a hilltop city. More like a mountaintop. Driving in and out of Siena can be intimidating. The roads at the base can be confusing. So can the drive up. The routes need to be learned. GPS can be unhelpful. Most first-time drivers to Siena come away with horror stories. I'm a crazy NYC driver (taxis are afraid of me), so I view driving challenges as an adventure. I think driving on the Amalfi Coast is super fun, especially in a convertible, so I'm not considered "normal" on most travel forums (and I enjoy that).
If you want to use Siena as a base to explore Tuscany with a car, it can be done, but the best in-and-out routes require study. You need to be extra careful how you plan your lodging. I find it easier to enter and exit the city from the south. If your lodging is not selected well, you can get stuck in serious traffic jams each time you attempt to depart and return to the city center. Parking is a serious obstacle within the walls of Siena, which requires visitors with a car an extra layer of lodging research.
Some people find it less stressful to use lodging in the countryside as a base. There are many lovely places within twenty minutes of Siena that offer a true Tuscan country experience. Parking and exploring with a car is much easier from lodging in the countryside. It's almost always free. My favorite parking lots in Siena's city center (parcheggi) are Il Campo and Stadio-Fortezza. Once you learn how to get to them and leave them, IMO, they offer the fastest way in and out of Siena. I have little-to-no patience for traffic jams. If you choose to rely on Siena's parking garages, you need to learn how to use them. The research never ends.