Yes, but most were bound for OR and went with the “Mormons,” probably for the safety factor. So there were only a fraction of non-LDS who used the handcarts.
Handcarts were used because oxen and wagons were hard to come by in the settlements that acted as staging areas. And as I understand it, there were some “freight trains” in the trek west, involving 20-mule teams. I mean, someone had to freight the goods to American settlers in CA...
It’s really extraordinary what people will do for trade!