I don't see where the traction in the corners is very valid. There is a big difference between suspending the bike (like MTB full suspension) and suspending the rider (Softride).

I have used a suspension seatpost on a hardtail MTB, again suspending the rider. It did nothing for cornering traction. It did make rough trails more comfortable when in the saddle.

Any suspension at all on a bicycle is going to cause a loss of pedaling efficiency. However, it can make the ride more comfortable. Like anything, there is always a tradeoff.