Here are my stock shapes that help with cooling and REALLY free float the barrel.
The main reason is that the space allows me to wrap the barrels with wet cloths. In the summer, our temps get up there and if I let it air cool, it would take about an hour. By wrapping with cold wet cloths, it only takes a few minutes even with these heavy barrels. The cooling is also even around the barrel.
Even if there is a "big" space, the stock is still an excellent insulator. I didn't like the idea of one half of the barrel "insulated". Maybe cause uneven barrel heating and more walking during the next string.
I put ice and water in a cooler and throw the clothes in that. Keeps the drinks cold too.
As to the guy thinking that the chamber is much cooler then the surface, I don't buy it. Yes, there will be a temp gradient across the steel but the difference when the outside gets hot, is minimal. The big differences would only occur if a coolant was running down the bore.
Remember the hot poker in the fire. The end you hold gets pretty hot. Steel transfers heat along its surface very evenly. Wood doesn't, but we don't use wood barrels.