All the tricycles I saw use chains. It looks far from being efficient comparing the regular bike:
Notice how long the chain is. They have to use 2 chain idlers. The back path is inside the curved pipe which is impossible to make frictionless.
At the same time, there are quite a lot of chainless bicycles:
Why not make the shaft longer, add one or two cardan joints, and put it right inside the trike's frame? What is the obstacle? Is it the weight or efficiency?
The chain return is under very light tension and running that across a low-friction plastic such as Delrin would not contribute significant load. Having the slack chain guided so close to the chain ring and rear sprocket means that it's less likely to jump off on rough terrain.
What size do you reckon the bevel gears are on the pedal-crank - drive-shaft and from the shaft to the hub gears? I'd say they're pretty fine and may not last that long. I would wonder about their efficiency too?
Almost all the trike's drivetrain is made of readily available components. It's not elegant but it's economical, reliable and fixable anywhere and fairly future-proof. Where will the other guy get parts in a few years time?
The nice thing about chains is that shifting gears is well developed and cost effective. I would imagine a shaft drive would be more complex and expensive while hardly any more efficient.
Shaft drives have additional losses due to gears.
I don't know why you think it "looks" inefficient, but mechanical principles work no matter what you think of them.
Motorcycle manufacturers sometimes use belt and shaft drives, but for aesthetics & maintenance, not efficiency.
The issue is torque. Go look at the driveshaft on a school bus. I used to make about the same torque as a school bus diesel when I was competitive. Less rpms, but similar torque, and I broke a couple cranks, too. I got about a week out of a chain and freewheel set. The propshaft would have to be pretty fat to survive.
