Siena does give plenty of options for day trips, including bus trips if you want to leave the car alone for a day.
What about Siena and Montepulciano as the two bases, by way of compromise? We did both on a prior trip and the experience is different enough that you'll enjoy the contrast.
Montepulciano is great to stroll around, very welcoming for wine tasting and has a nice cosy feel. Siena is bigger, and whilst the Piazza del Campo and main pedestrianised streets around it are touristy, the city is more than that. Both are hilly, with Montepulciano shorter but more brutal, whilst Siena can hide the rise and fall a little more.
Would you consider taking a taxi from the airport to Montepulciano, having a couple of days there to get over the flight, then hire a car to go to somewhere near Siena for the remaining 4 days?
FWIW we stayed in apartments that would be no more than a €10 taxi ride from Siena, and even despite my love for local transport, this made a nice compromise of being outside, but with no concern about getting a taxi back. I can't say I'd recommend the apartments - a little too modern / purpose built / campsite for our tastes, but I daresay there are better alternatives within ~5 km of the city.