No. Divers lose much more heat through conduction than respiration. And the emergency reserve gas only lasted a few minutes so once that was exhausted he wasn't breathing at all.

A rebreather helps a little because the gas stays warmer but it isn't a huge factor. I've been using open circuit trimix for 20+ years in the cold waters off of California. It's fine as long as you have a separate tank for drysuit inflation. Divers may "feel" colder when breathing trimix but this is an illusion due to the reduction in gas narcosis: one of the effects of narcosis is to make you feel warmer even when you're hypothermic. Helium is an excellent thermal conductor but the heat capacity is actually lower than nitrogen so all else being equal you're not losing body heat any faster by breathing trimix.

There's lots of mythology in diving. Even a lot of technical diving instructors seem to be unclear on the basics.