Those trees grow wild everywhere around here and that is how they always react when cut down, there is a lot of life in them. You can cut away all but the main stem if you like, that will encourage growth of it. Suckers will continue to come every year though, once cut down they will never voluntarily grow singel stemmed any more.

Same here. Seedlings pop up all over.
Like Huggorm said, they sucker if cut back hard, otherwise it's not a problem.
There are some very ornamental varieties. Some with white or yellow berries, rather than the common orange or red.
Mike