Wrong. Reducing throttle in cold air assures a lean condition that you wouldn’t see in warmer temps. When it’s cold out? Full throttle is more important than ever. Turning the prop down a little? Look at a prop tip speed calculator and do your own math.
You're not wrong that in cold weather you are seeing a lean condition that you wouldn't see in warmer temps but I do believe you are wrong about how you would go about fixing it. Consider that the mixture is already set to full rich on such days, you are physically incapable of increasing the amount of fuel you add to the engine. The only thing still within your control at that point is the amount of air entering the engine by adjusting the throttle. Reducing the throttle in this condition brings the engine back under control and closer to the ideal fuel-air mixture of 15 to 1.
Throttle controls the amount of air entering the engine not the amount of fuel. If you want to change the amount of fuel entering the engine you change the mixture not the throttle.
While we normally think about engine limitations from a fuel perspective, the normal limiting factor in engine combustion, especially the rate, is normally the presence of air, not fuel. When we adjust the mixture, particularly in fuel injected engines but this also applies to carbuerated engines, we are setting a quantity of fuel to be added to the air mixture; that quantity remains the same for every revolution of the engine regardless of throttle setting. When we adjust the throttle, we are adjusting the amount of air the engine is allowed to intake.
Going up in altitude we set the throttle to full because the air is less dense so we need to intake more of it and we then use the mixture to smooth out the engine as continuing to climb results in increasingly less dense air that the throttle can no longer compensate for thereby causing the engine to get too rich and close to flooding. Going down in altitude is fairly similar, its not quite an exact mirror as you have more options on how to obtain a given power output but its close enough.
On the ground, if we set the mixture too low, we end up with too little fuel for the amount of air and we starve the engine of fuel until it stalls
If we set the mixture very high on a low air density/high altitude density day, we end up with too much fuel for the amount of air and we starve the engine of air until it stalls. Since we normally look at combustion as a factor of fuel not air, we call this "flooding" the engine. Pretty much every "flooded engine" start procedure I've seen is the same...
Set mixture to idle cutoff, throttle to full. You are literally setting the throttle to allow full air flow into the engine while setting the mixture control to allow no fuel into the engine. Sometimes all you end up doing is burning off the fuel and having to start anew with a "non-flooded" engine but often times you'll successfully get the engine to turn over and run for a moment, until the fact that you aren't adding fuel to a full open throttle eventually starves the engine for fuel and it quits again.
So what does that mean on a cold weather day?
Well you have a static, unincreasable (full rich) amount of fuel that you are putting into the engine. The engine is producing more power than desired though because this full-rich mixture is actually fairly lean as a result of the high-density air. In order to correct for this high-density air and increased power output, you need to decrease the power output by decreasing the amount of air the engine intakes by decreasing the throttle setting, stabilizing the fuel-to-air mixture/ratio and limiting power output.
Now in most cases with a normally aspirated engine, you probably wont run into an issue... For each 5 degrees below standard, you get about 1% increase in engine output so it'd take about -40C day at sealevel to reach power outputs above 110%. So in most cases (and for most of us) you're probably ok to take off in your normally aspirated engine aircraft with full power so long as the engine is properly warmed (dont demand 110% of your engine until your engine is fully ready to give it to you) and you dont remain at that power setting for long after takeoff.
For supercharged and turbine engines however, the FAA advises that you use caution not to overboost and that you refer to your aircraft/engine POH to determine the appropriate power settings to use for the given pressure altitude and ambient temperature.
Lastly, I'd like to note that I did not advise using the governor/manifold to adjust prop speed but rather power; sorry if that was unclear. In fact, in a lot of ways using manifold to adjust prop speed would be much much worse since it is possible you'd be overpowering your engine without even knowing it since your manifold pressure and RPM's would not be above redline even though your engine is producing significantly more power. At full forward prop though, the prop RPM is controlled through the throttle just like it would be in a fixed pitch airplane. If you are at or above redline or desired/target RPM at full power with full forward prop (which would be standard for takeoff), you would reduce power to reach targets or prevent over-revving and damaging the engine. Should you take off in such a condition is a different matter.
