Yes they can tow a vehicle with broken axle or broken springs.
It is called a flatbed trailer. It also works when there it a accident so sever that the vehicle can not be towed in the normal manner.
Flatbed tow trucks are so common today that we are not used to seeing the sling type tow trucks anymore, with the two wheel dolly's that where so common place in the 60's and 70's. Basically the dolly has two wheels on each side, the car is lifted by the tow truck, the dolly assembled from both sides of the car and slid together under one side of the car, then the tow truck drops 1/2 on the dolly, goes around to the other side, picks up that end of the car, and then tows it where ever it needs to go. This sort of dolly could have transported your dis-abled trailer as well.
I really don't think the towing company will show up expecting to find a trailer with a broken spring and say "There is nothing we can do to ever get this vehicle off the side of the highway" - if they did, it might be a teenager driver who has no clue on what to do. Even a tractor trailer on it's side with the sidewalls torn apart can be moved to a repair facility or impound yard. They do not just leave them where they crashed forever, normally they are moved within 2 hours. . .
Look at the red dolly in this picture. It is a 1970 tow truck.

Fred.