Kinda funny, but I do most of my "cutting at the muzzle" in my loading blocks. I make the blocks the same thickness as the ball diameter, then seat them with the ball protruding just far enough out the bottom for an easy feel when they're centered over the muzzle. That leaves the top of the ball somewhere between 1/8" and 3/16" below the top of the block, where I cut the patch. When I've emptied a block at the range, I usually just take a few moments to reload it this way, rather than seating the ball in the bore, then cutting the patch. Handy.
Most of my range blocks hold 6 balls, but my big game hunting blocks hold only 3. I'm even thinking about going to 2. If that's not enough on a big game hunt, a truck load of balls isn't going to be enough either. My small caliber snowshoe hare blocks hold 6 too, and I usually carry 2 of those. And I'm happy to report that I've had to refill those blocks in the field!