I think it depends on how you envision your trip going. It's perfectly fair to want a portion of your trip to be relaxed without so much get-up-and-go. If you've already visited Florence, dislike crowds and noise, aren't super into art and history, have mobility issues or just find great joy in relaxing with a glass of wine overlooking rolling hills for many hours a day, then Siena would be a great place. My vision of a great trip is getting out and seeing beautiful art & historical buildings, walking streets lined with countless shops and outrageous eateries, and soaking up a variety of churches, monasteries, and open-air markets. We love walking miles and miles a day so we can eat, eat, eat. My teen and I climbed the campanile (bell tower) and cupola (dome) within an hour of each other, walked to our hotel in the Oltrarno, all the way up to San Miniato for evening vespers, Palazzo Vecchio area for dinner, then back to our hotel near Santo Spirito...easy day. Siena can be "seen" in a day, and without a car, unless you're staying right near the restaurants you want to patronize, you have to take taxis or mess with a bus schedule to get anywhere else. To get the beautiful views, we would have had to stay outside of town, and it would be more of a nuisance to get anywhere (and always a 2.5- to 3.5-hr round trip bus ride to see Florence). Our family (plus the in-laws) is returning to Tuscany this March, and we had to make this exact decision. We initially booked 5 days in Rome, 3 days in Florence. Instead of tacking on 3 extra days in Siena, we got a different (cheaper) VRBO in Florence from which we will make our base for day trips to Pisa and Siena and seeing more of the city we left uncovered.
We stayed in the Oltrarno last March, and it was still really close to all the attractions. We did 3 days and still could not see it all. This time we're staying near the Duomo and will enjoy being even closer to our room to put our feet up, use the restroom (without paying to eat somewhere), refill our water bottles, and charge our phones from all of the pictures we took, directions and hours we needed to look up, digital menus, digital ticketing, etc that drained our battery. There was also far more to do during the day and so much more to do until all hours of the night (and many free churches and museums with a nominal fee). I could spend a month in Florence and never get tired of it.
If you want to save on lodging in Florence, pick an attic apartment on VRBO without an elevator. For 6 people with 3 bedrooms, 2 baths (private loggia & slight view of Duomo), we are spending less than $250/night with taxes and fees. Our first 3 days in Florence we're on the 2nd story with a lift and are spending $580/night. On our last visit to Florence, we paid $250/night for 2 people (1 noisy hotel room on a piazza--will never make that mistake again). In Siena, we found rooms for $110/night for 2 people, $150 for 4 at lovely wine estates with great views on the property.
