I don't think the current state could be described as a "race".
NVIDIA makes good stuff and great drivers but charges a fortune (+$1500 for a high end GPU?!) AMD has made space heaters that happen to have video-out ports and their cards are useless for compute work because so many projects support CUDA and nothing else. Their current lineup cheats benchmarks by overclocking the cards and thermally throttling shortly after the time it takes to run a typical benchmark.
Edit: since I'm now being 'corrected': look at the TDP for same-generation, same-product-tier NVIDIA and AMD cards. AMD's cards have always used more power for similar performance.
People think certain GPUs "run hot" because
- they buy the cheapest GPU they can which means the heatsink and fans are terrible
- they don't adjust the stock fan curve (most GPUs these days don't even turn on the fans until they hit 60c and only max out the fans when the GPU die temperature is very hot because people care mostly about noise
- they don't have a well-ventilated case (note I said well-ventilated, not "stuffed with fans)
Buy a properly ventilated case, a mid-or-high-tier card from a reputable manufacturer, and adjust the fan curve. Find the temperature the card sits at during light desktop use and set the fans to kick in slightly above that. Then force the fans to run at full speed and benchmark, seeing what the die temperature maxxes out at. Set the fan curve to 100% at slightly below that temp.