I found a machine that will sew bags! It's a Sailrite Ultrafeed LSZ. I'd never heard of this brand before, but I saw it on Adam Savage's Youtube Channel so I ordered one! This was the third machine I've purchased because I want to make bags.
It took a bit of time to get it all set up. (Sorry, I haven't yet managed to get a good photo.) The machine is crazy heavy, over 50 pounds, and I don't trust it won't break my sewing table if I put it in the cutout, so I am still waiting for the new sewing table to set it up in a permanent location.
I decided to give it a test drive by sewing a new purse. I spent a couple of days measuring my favorite purse and reverse-engineering the pattern. I decided to cut up some old, dead jeans for the fabric.
I made a few minor changes from the original purse but I think it turned out really well for the first try. And the sewing machine performed great. Now that I've figured it out, I think I'll make another in purple, my favorite color.