I have used airless tires, and bought them for the same reason: being driven crazy by flats, almost always on the rear tire. So I put a solid tire on the rear. I commuted that way for about 6 months, and as you might imagine, it is a foolproof solution to the problem of flats. I didn't notice a big difference in rolling resistance, but I may not be a very good judge of that. It was nice not to have any flats.
But I gave up on the solid tires after about a year for two reasons. The solid tires seemed to wear out pretty fast, and though giving a good drip on a dry road, they seemed more prone to slip on wet or frosty pavement.
I helped my flat problem with kevlar rear tire, a bigger tire width (40), and h a wee bit of tread on it so I have a little more rubber for thorns, wire and glass to have to get through. The flat I had (yesterday!) was the first in almost a year of 100 mile per week of commuting on urban roads.
So, I prefer my new solution, but solid tires are definitely a brute force solution to flats, and on dry pavement, they work pretty well.
Chero