Also, not sure how old your kids are, but it seems that your initial pricing comparison was for one cruise cabin. If you do an ABD - you would normally get two rooms for 4 people. They indicate that they can't guarantee connecting rooms. We had 3 different hotels in the Italy trip we did, 2 were connecting, one was next door but not connecting.
We thought that having two bathrooms and more spread out space (European hotel rooms tend to be smaller than US rooms) was great for our family (with two teenage girls). Also - having real beds after long hot days was also helpful for keeping them happy and rested (versus a pull out type bed situation).
Just another thing to consider when doing your value comparison.
Oh, and we had air conditioning in our hotels. Maybe not quite the same as US, but generally comfortable.
And the guides do most of the logistical work of travel - checking you in and out of hotels, moving the luggage, getting you to your transportation (such as if you use a train, etc.). I didn't find any of this stressful when I was traveling with ADB (but can find it stressful when doing it in Europe normally).
I think that you will have a great vacation either way - but it really depends on what sort of experience you want.
(As an aside - I wish that they still had those ABD ad-ons for the European
DCL cruises. Seems like the best way to have a great port experience on a big cruise, for people used to ABD touring.)