What really happens when you go towards peak EGT is you're assuring a
pretty efficient burn inside the cylinders, the exhaust contains little
unburnt fuel and little oxygen. If you play a little with the mixture
when prop and throttle are firewalled you'll probably pick up some RPM,
even at low altitudes. It's especially noticable on hot days.

In cruise with constant speed props the prop pitch changes a little,
taking a healthier bite of the air so as to use the additional power.
CHT is a simple way of measuring what's going on -- it's telling you,
as you go from rich to lean, when you're getting the hottest -- call
that best -- internal temps. Since the fuel (I was taught) helps in
cooling, you want not to overlean so as to save the valves. That
statement probably deserves some research -- why not run schiometric
(I'm sure I mis spelled that) combustion? Does anyone know for sure?

For that matter, if you do get peak power at max EGT, why run on the
rich side? Wouldn't it make sense in terms of fuel economy to run on
the lean side a bit?

I know the dogma, what I'm interested in is the justification. Any
powerplant engineers out there?