The hand cart isn't in this movie for very long, but Mel Brooks uses it to make a point about how little value the white bosses of the Old West put on the lives of black people. You got to hand it to Mel Brooks—this could be an excruciatingly dismal symbol, but this director makes it funny. (Even if it's more like a "heh" than a belly laugh.)
Even when Bart and his buddy Charlie fall into some quicksand, their white boss Taggart is much more concerned with saving the hand-cart they were driving, as he says,
"Dang, that was lucky. Dog-gone near lost a $400 hand cart."
After this kind of crummy treatment, it's little wonder why Bart picks up a shovel and smashes Taggart in the back of the head. (Which makes us laugh for real, as well as doing a little fist-pump.)