I agree that if your side-cutting can opener is completely cutting off the bead of the can and leaving it floppy, then either that can opener isn't designed properly or might possibly just be made a bit sloppily. Or possibly you aren't using it as the manufacturer intended, such as tilting it so that it locks onto the can under the bead.

I don't recall what brand my manual can opener is, and there aren't any markings on it, but it reliably splits the top edge of the bead of every can. The can is still sturdy, no sharp edges, and no lid falling into the food. It's great!