Correct, it is well known that in total many more people die from cold than from heat. Obviously the numbers vary per season and location. My point is; we have a starting figure and a trend for both higher and lower than optimal temperatures.
The Nature paper also only shows the right hand side of the story in figure 4. If you honestly think mortality is the problem, there should be a red line somewhere below 0C as well. Assymmetrically linking only 'bad stuff' to 'higher temperature' may be counterproductive if we want to come to a smarter policy (yes, I work in government, and I think it could be done...).