I seem to recall some patches that had a coating on the side away from the tube, with a sort of release paper over it as received. The tube would be scuffed up with the little grater-looking thing from the kit (often the thin steel lid of the cardboard tube container), and glue applied to both tube and patch; after it started to set up, they were put together, the release paper removed, and then the clamp [or, "Cast Arn Thingy"] used to hold everything in place; the coating beneath the release paper was what got lit on fire. Having only a few small contact points, the legs of the clamp, gave free access to air. It burned, but not a big flame; afterward, a sort of traveling front would move around on the surface to complete the burning (like what you see if you burn a newspaper consisting of multiple sheets in contact). In other words, it WASN'T the glue that was being lit on fire, but a special coating on the opposite surface of the patch.
There may have been multiple kinds of patches, each avoiding other manufacturers' patents; I would only have seen the kind my dad used.