CAMom, maybe I'm misunderstanding your point or maybe schools where you are are very different from schools where I am.
I work in a middle school and have been in a high school before in the US. This is how all the teacher periods are structured and students have 6-8 teachers in a day, depending on how the school is structured. If we used this model more in our elementary schools, you wouldn't have to hire more teachers, you'd simply have to have the students rotate classes more often. For example:
You have 5 groups of 20 students. You would have 6 teachers. At any given time, all 5 teachers are teaching their subject matter. Then they swap around. It works best if you have an area or two that do not necessarily need to be leveled- like PE or technology. It does take some work to make the scheduling come together but it's completely possible. We do it every year with 3 grades of students who need 6 different levels of math and 5 levels of Language Arts. It's not always a perfect fit but it's far better than saying "You're in 2nd grade, so you get 2nd grade math. End of discussion."
