Stainless steel does not conduct heat better than alloy steels, quite the opposite. Thermal conductivities of carbon/alloy steels are typically around 90 W/mK, compare to around 30W/mK for most stainless steels.
I have never seen any real, well researched facts about stainless steel in gun barrels. There is tons of internet blather and gun shop "facts", but not a word I would take seriously. The only things that are true, bordering on self-evident:
- They resist field conditions better, as they won't rust or corrode to any appreciable extent
- The grades used in barrel making (usually 416) are easy to machine, saving costs for the manufacturer, and/or allowing him to easily hold tight tolerances.