I think it's an interesting idea. I'd draw a comparison with unicycling. We usually don't have changeable gears on unicycles, so our nearest equivalent is changing the length of the cranks: shorter cranks = more speed but less torque; longer cranks more torque but slower. Better quality unicycles have cranks drilled with 2 or even 3 holes. Therefore, instead of changing the cranks half way through a ride, you can move the pedals to the "short position" or the "long position". It takes a minute or two, job done, and it means you can ride fast along the road, then change to a more torquey "gear" when you reach the off road section. The idea is simple, robust, fairly cheap (a geared unicycle hub is incredibly expensive) and sounds wonderful.

In practice, I do not think I have ever swapped the pedals from one hole to another part way through a ride. However, I do choose the best position before my intended ride.

On my fixed bike, I have a flip flop hub and have never once flipped the wheel during a ride. However, I am aware that sometimes I choose my route within the limitations of the ratio that I have available. Maybe this idea would be a good compromise.

A "dinglespeed" with freewheel would have no appeal to me. I typically ride my 2 x 10 for many miles at a time without changing gear and treat it as a "selectable single speed." However, for fixed, the 2 x 2 approach would be worth considering.