I've been reluctant to chime in on this for fear I am missing something special other people are seeing. I have mounted everything from bicycle tires to car to motorcycle to tractor tires by hand (without bars)and the rule is the same. The opposite side tire bead(s) must be kept in the center of the opposite side of the wheel. This is because the diameter of the wheel in the center is less than the diameter at the edges. The tube must be nearly flat.

It's like putting one of those "Do Not Disturb" tags on a door knob. They work easiest if the leading edge of the whole goes over one edge of the knob into the narrow space then the opposite edge is looped over the other side.

There are enough stories floating around about "this tire" or "that wheel" that is particularly hard, so my mind is still open; I just haven't seen one of them yet.