"Why aren't 4 barrel sets popular in competition?"
The biggest part of the equation is weight. The average weight of a skeet gun shot by a top shooter is 10lb, some are more than that, and very few are lighter. Almost 1lb of this weight is from the subgauge tubes, which don't exist when shooting a 4 barrel set.
Another factor is disparity in weight and balance from barrel to barrel. Very few 4 barrel sets will feel and swing the same across all 4 barrels. I believe Krieghoff is getting them a LOT closer no a days on their barrel sets, but they also get around $5k for a set of barrels too. Off the top of my head, I believe they are the last mfg offering a 4 barrel set on a non-custom basis.
And in that vein, cost is a HUGE factor. Figure a new std grade K80 4 barrel set is going to run you around $25k. Or a 1 barrel set with subgauge tubes will be about $11k.
Final factor is perception. The view over the barrels on a 410 barrel set is dramatically different than the same view over a 12ga barrel.
With subgauge tubes, you have the same weight, balance, look, and feel when shooting 20ga, 28ga and 410, and if you get a weighted barrel for 12ga, for that gauge as well. There really is NO advantage to a 4 barrel set, and NUMEROUS advantages to subgauge tubes.