If I remember right the original grillz straps are sewn on to those clips so you can cut them off and then the sort of ridge that is on standard straps (including that JT one) can be pushed through and then when you try to pull it back it lies flat and can't fall out.
I've never replaced the original grillz strap but I have installed a JT strap on top of it before by pushing it through in the same way.
EDIT: you can double strap grillz in the exact same way as JT proflex so it must work with those kind of straps.