Agriturismo in Tuscany

La Fonte near Pienza gets rave reviews. Car definitely needed.

https://www.wanderingitaly.com/maps/valdorcia.html

My husband and I have stayed at the Relais Villa Grazianella, an agriturismo outside Montepulciano and loved it. It is a hotel and winery with a marvelous restaurant. There is also a lovely swimming pool which I think your boys would like. We were there in September, so I don't know whether it has AC, but their website should say.
There are many small towns and the Le Foce Gardens within easy driving distance.

Take a look at this property. They have many experiences for children.

https://www.theisabellaexperience.com/experience/

We found the Le Foce Gardens to be a big letdown.

tips from a local:
my recommendations : Podere Benintendi Certaldo(Fi)
Guardastelle San Gimignano(SI)
are good bases for visit the central Tuscany
you can see rewiews on Google,Tripadvisor ecc.
Enjoy

I’ve always had good luck with Sawdays… you can sort and refine by a range of amenities and view by map. They are usually very specific about accommodations. They have almost 100 options in Tuscany. Once I find a place I like I’ll cross reference on TripAdvisor. Terre di Nano near Montepulciano is absolutely fabulous but they stay pretty booked. They have a range of cottages and apartments on the property, a great pool, and a wonderful chef! Check with them. Also you will definitely want a car.

Sawdays.com
http://www.terredinano.com/

I highly recommend Sant' Antonio Country Resort outside of Montepulciano for a week stay. We LOVED it.

I don't have one agriturismo to suggest which meets all of your requirements, but I can give you the best websites where you can do a search yourself, and maybe place the appropriate filters to find what you are looking for.

BEST WEBSITES TO SEARCH AGRITURISMO ACCOMMODATIONS:
https://www.agriturismi.it/en
https://www.agriturismitaliani.it/en/
http://en.agriturismo.com/index.asp?idLingua=2&id=
https://www.agriturismo.it/en/
The websites will list the description of the properties and the services available (like A/C, swimming pool, restaurant, etc.). Some websites allow you to filter the type of services too.

LOCATION CHOICE:
In the region of Tuscany there are 10 provinces (provinces are similar in size and population to US counties, a region, like Tuscany, is more like a small State, for example Tuscany is a bit larger than Massachusetts). The 10 provinces are: Florence, Siena, Arezzo, Prato, Pistoia, Lucca, Massa-Carrara, Pisa, Livorno, Grosseto. In any of the websites above, first select the Region (Tuscany) and within the properties listed for that region, the property listed will list the city/town/municipality and, in parenthesis, the province.
Since you have never visited Tuscany before, you probably want to be in the area of Tuscany that offers that most famous places to visit. That area is, in my opinion, the Province of Siena, or the southern part of the Province of Florence (basically at the border with the province of Siena).
Once you have zeroed in on the property that fits your needs in that province, make sure you also plot it on the exact location within the province. Some properties are located in isolated pieces of land that are not easily reachable from the main highways. This is important because if you want to take day trips to your intended destinations you may not like spending a lot of time on a curvy country road, just to get to the main highways. The best way to check is to plot the address on Google maps, then test the distances from that property to the towns you want to visit, that would give you an idea of the driving time. Also you can use Google StreetView to see what kind of road you have to drive near the property. Some require driving long unpaved driveways from the main road.
As I mentioned, the best areas are near Siena or between Florence and Siena.

Give it a try and good luck!

All of this is extremely helpful. Thank you all so much!