By definition, EGT is lower when richer than peak EGT.
By definition, EGT is also lower when
leaner than peak EGT.
Full rich generally gives lower CHT.
Full lean always gives lower CHT. Any mixture that's leaner than about 50 degrees F ROP will also give a lower CHT than at 50 degrees ROP.
So, taking away fuel can also cool, and adding fuel can also heat, depending on where you are on the lean curve before you start taking the fuel away or adding the fuel.
So I don't understand the claim that fuel doesn't cool. If richer mixture doesn't cool, where does the heat go with all these cooler temperatures?
If you start at peak EGT, and add fuel (make the mixture richer) so that the EGT goes down by about 50 degrees F, your CHT will go up. So in this example, adding fuel doesn't cool.
If you start at 50 degrees F lean of peak, and add fuel (make the mixture richer) so that the EGT peaks and then goes down 50 degrees F on the rich side of peak, your CHT will go up. Lots. So in this example, adding fuel still doesn't cool.
Making your mixture increasingly rich from around 50 degrees F ROP will result in lowering of CHTs, because all other things being constant, as the mixture is made richer from around 50 degrees F ROP the peak pressure point in the combustion cycle becomes later, and therefore the peak pressure becomes lower. (Using a higher RPM will help as well.)