Always size the needle to the thread that you’re using, not to the fiber being sewn, and you’ll have the best luck. Larger needles need more punching power.
My wife sewed many, many embroidered patches on our leather vests with a plastic-geared JC Penney 7057. The reason that she could do it, was that she used a size 11 universal point needle and Tex27 serger thread. She tried leather needles and kept breaking them. It doesn’t take extremely strong stitches to simply hold a patch on, and I know that you’re stitching for strength making a bag, so it’s an apples-oranges comparison, but a household machine can punch a thin needle through some fairly thick and dense stuff. They’re all going to work overly hard punching a size 21 needle through very much thickness. Just IMHO.
CD in Oklahoma