I do notice a difference, and first started doing it in the late fall when I would start wearing thin gloves, and the tape did help. I also have experience from getting really cold hands and fingers from handling tripods and whatnot in the cold, and in extreme cold, even with thicker gloves I don't like touching bare metal, so another reason to do this to my cold weather bikes, commuter, winter commuter and fatbike. Plus I have the hockey tape, so its around and doesnt take long to do.
You are Canadian, of course you have hockey tape.
Several years ago you could order bar tape from Asia on Ebay for an extremely low price, they called it cork but at that price I assume it has no cork in it. I bought a lifetime supply. It had no adhesive, you needed to buy some two way tape at an office supply store for it. Have that on all of my drop bar bikes.
One of the joints on my thumbs get really sore where it contacts a kayak paddle, so I started using padded handlebar tape on kayak paddles, not only does it feel much better on my thumb joints, but in cold wet conditions, it has much better grip. I have my second paddle strapped on the foredeck on my kayak in the photo, the bar tape has a yellow and black pattern.
I still have handlebar tape on the brake levers on one of my bikes. This is my medium duty touring bike. The tape was black at one time, but a lot of sun turned it gray.