It's easier to clean up though.
I'm still not following you.
However, if you draw the toilet paper out with your "clean" hand, then switch, grabbing the end of the roll with your "dirty" hand and tearing it while stabilizing the roll with your "clean" hand, the roll should remain clean.
Is there really a problem with people getting the toilet paper dirty? The only problem I've ever had is when someone unrolls too much and it's touching the floor. I simply tear off the "contaminated" portion and throw it away.