mike,
My preference would be to leave Rome in the Itinerary at the end of the trip, not only because it has a lot of iconic sites that would be of interest to the kids, but it's also a convenient location for the outbound flight.
I would also suggest skipping Verona, especially at that time of year as that's getting into Opera season and rooms will be in high demand and probably expensive. However there is one attraction in that part of Italy that the kids would probably enjoy - https://www.gardaland.it/ .
I respectfully differ from the others here, and would not suggest adding Venice to the list. In the interest of full disclosure, I'm not as enamored with Venice as many in the group. While it's very unique and deserves at least one visit, seeing the sights there during the daytime with the current overcrowding situation may not be a pleasant experience, especially for the kids.
As you may be aware, for driving in Italy, each driver listed on the rental form must have the compulsory International Driver's Permit, which is used in conjunction with your home D.L. These are valid for one year, and easily obtained at any CAA/AAA office (two Passport-sized photos required, which may be provided by the issuing office). Failure to produce an IDP if requested can result in fines on the spot! Have a look at https://it.usembassy.gov/u-s-citizen-services/local-resources-of-u-s-citizens/transportation-driving/ for more information on I.D.P's and driving in Italy.
You may also want to have a look at some of the other posts here concerning the dreaded Zona Traffico Limitato (limited traffic) areas that are becoming increasingly prevalent in many Italian towns & cities especially Florence, which is almost saturated with automated ZTL cameras. EACH PASS through one of the automated Cameras will result in a €100+ ticket, which you won't know about until several months after you return home! This website provides more information - https://www.italybeyondtheobvious.com/dont-mess-with-ztl-zones .
How many days do you have in total? Keep in mind that you'll lose a day on either end of the trip for flight times. You'll arrive in Europe the day after you arrive and the last day will be spent on the flight home.
If I were arranging this trip, I might try something like this.....
D1 / N0 - Flight to Zürich
D2 / N1 - Arrive Zürich, train to Mürren
D3 / N2 - Mürren & area
D4 / N3 - Mürren & area
D5 / N4 - Mürren & area
D6 / N5 - Mürren & area
D7 / N1 - Train to Florence (it will mean an early start and a long day, but will fit well with the rest of the trip - the kids can sleep on the train - if you leave Lauterbrunnen by 07:00, you can be in Florence by 13:00 and still have a half day for touring).
D8 / N2 - Florence, main sights
D9 / N1 - Rent car, travel to Agriturismo in the Siena area (hopefully the Agriturismo has a pool)
D10 / N2 - Agriturismo, touring
D11 / N3 - Agriturismo, touring
D12 / N1 - Either return car in Siena or drive to location in Rome and return car somewhere outside the ZTL areas - you should still have at least a few hours for touring.
D13 / N2 - Rome, touring
D14 / N3 - Rome, touring
D15 / N - Outbound flight from Rome
There are many ways this could be structured and this is only one suggestion. This saves two travel days. You won't need a full day for travel from Florence to Rome, as the trip is only 90 minutes. While it only allows one night in Florence, you'll have a short time to see a beautiful city. You could also add one night in Florence and decrease one night in the Agriturismo. Florence is a great location to use as a home base and take day trips to Lucca, Siena, Pisa, the Cinque Terre or even Venice.
I'd highly recommend packing along a copy of the RS Italy 2019 guidebook, as it will be an enormous help in planning sightseeing, hotels, transportation, etc.
Good luck with your planning!