The cross draw can be done with most holsters. It can also be done without sweeping your body. Care must be taken to prevent sweeping your left arm (assuming you're right handed.
To draw with the left hand simply twist your hand so its flat and facing out or away from the body, slide it down where you can get a grip on the gun, then pull the gun twisting it as it comes out of the holster. It doesn't cross the body.
KEEP YOUR FINGER OFF THE TRIGGER.
Like anything else it takes practice. For defense classes I have a set of blue plastic training guns, but an empty gun works if one is careful to make sure its empty.
Well, you know an ax just has one handle too. But I use two hands using one.
That is quite true, but when I'm using an ax I'm not normally doing anything else, both hands are free to grip the ax handle.
I'm not holding a flashlight, I'm not opening doors, I'm not poking a mirror around the corner. I'm not pulling or pushing a family member out of the way.
Can't say the same thing for a pistol/revolver.