Deacon it's hard to tell from the pics..is your brake the type where the wire splits into 2 and each piece runs to it's own side? If yes, it is a cantilever brake and I can see why you don't like it (I am extremely prejudice against cantilever brakes.)
This may be a bit advanced or even a nutty idea, but here goes my brain working in overdrive again....for the drive wheel:
Basically what you would do is make your drive wheel have the reverse tread of the bike tire. So to do this you would need an identical spare tire that you can cut up...Put a clay like material on the drive peg, all the way around. Then take your spare tire piece and wrap it around the peg, imprinting the tread, only once. Then take it off and let it dry. Then, coat with a very strong resin - polyester resin (for fiberglass) would work, as would most any other epoxy product that's not too thick. You could just dip the peg a few times.
When it's dry, you would have a peg that "locks" in with the tread of the tire.
I guess another way would be to coat in the clay like previously stated, but then lower the engine to contact the tire, and slowly rotate the tire to imprint the tread on the peg.
dunno, just a crazy rambling thought. Excuse me if I'm a little too "mad scientist" for your thread