I’ve made use of game carts, 4 wheelers and UTV’s, mini-trucks, actual trucks and SUV’s, drag sleds, simple ropes, improvised travois’, load-hauling packs, and American muscle to transport game from the field.
I’ve pushed wheelbarrows for hundreds of miles in my life, and nothing on God’s Green Earth would make me believe it to be a worthy tool for game transport.
I’ll mention with confidence and experience which our OP lacks, to choose to use pneumatic tires on a game cart is to choose to not use a game cart. The hassle and time lost in patching, repairing, or replacing tubes or tires will far outweigh the time and effort to simply drag the animal on the bare rim or on the ground.
I tend to find a travois quite pleasant to pull if I can’t get quarters or complete animal on my pack. Otherwise, a solid tire game cart, especially pulled behind my 4 wheeler is my favored method.