Dismounting the old tire (manually) sucks more than installing a brand new one. Sidewalls get stiff over time. Sometimes the beads get nicked, too, so you have to worry about them sealing on the new rim.
I second the HF mounter idea, spend a few hours with that and sixty bucks will never seem so cheap!
I second the HF mounter idea, spend a few hours with that and sixty bucks will never seem so cheap!