I had three paper mulberry trees in my yard (all males that don't set fruit). They are incredibly invasive and their root system goes every where. I had to dig a trench of a new water line to my house and it was incredible how many of these roots I had to rip out.
I chopped down all my paper mulberry trees and every time a sucker pops up I rip it out of the ground, you don't have to take the entire root system out, just rip out the suckers and eventually the root system will die of starvation. I suppose there is some type of spray you could use, but I don't use any sprays in my yard, including the so-called organic sprays.
I had logs of this tree re-sprout, so yes it's easy to get a cutting and replant. However, unless you're getting fruit these trees are just not worth it to me, too invasive and they are subject to serious insect infestation. All my trees had serious insect damage.
P.S. Even if I was getting fruit, I think I'd still chop them down.