How can I fix my ineffective, slipping cabinet hinges without getting new cabinets?
It looks like the part of the cabinet that is gripped by the hinge has a reverse vee shape so the hinge should not be relying on friction to stay in.   You should look more closely at wear on the wood or the hinge that is preventing the hinge from taking advantage of this profile.  Your pictures are great but I can't quite see what might be wrong.  Open a good one to compare.
I may be wrong, that may just be deformity from moisture.  I've never seen a hinge like this before.  It's clever.  I'm rooting for it.