I used to grow mulberries from cuttings all the time, no rooting hormone or anything, just poked them into a pot of soil. That was on the Big Island of Hawaii, though; very rainy a lot of the time, and warm and humid all the time. In a drier, temperate climate you might want to use rooting hormone, and I am betting a warm, humid greenhouse-type environment (or the closest to that you could create) would be where you would have the best luck getting them to root. I don't have experience rooting plants in temperate climates (though I just moved to one, so I'm about to learn!), but just wanted to say that yes, these guys do want to root, and you just have to figure out the best way to do it. These instructions look very thorough:
https://www.growingmulberry.org/propagation.
I bought a few different mulberry trees and planted them at my new place. When they are big enough, I definitely plan on propagating more with cuttings, both to plant here and to pot up and sell.