Elms are notorious for budding from the cambium at a cut site so what you have is normal. I normally try to make elm cuttings the length I need for the first trunk section. One of the new top buds becomes a first branch and another will continue the trunk line. For broom style make the top of the cutting where you want the branching to start.
Hard pruning the tops can sometimes trigger buds further down the stem but the more you want it to happen the less likely it seems to be
Cutting through the bark and cambium can sometimes trigger buds just below the cut so you could try a horizontal cut just above the spot you want the new buds.
I have resorted to grafting on a couple of larger cork bark Chinese elms that needed lower branches but just wouldn't sprout buds no matter what I tried.