As I believe you know engines are specifically run at richer than best power mixtures when the mixture is full in (below 3000 feet).
Yes. The reason for this is not for actual smooth running, but rather for cooling. In fact, when you run full throttle and full rich, you actually reduce your power available. How? Well, you are dumping extra fuel into the cylinders. That extra fuel cannot burn, because there is not enough oxygen for it to combine with. However, the heat of combustion causes this liquid fuel to evaporate. The process of evaporation absorbs heat from the burning mixture in the engine, helping keep the engine cool. Heat, and the accompanying expansion of gases in the cylinder, is what makes our engine go. Less heat=less power. The decrease in power by running very rich is considered an acceptable tradeoff for not melting our flimsy aluminum cylinder heads.
So we start are engine and decide at 2000 feet density altitude we should lean are mixture. We lean for best power and start to accelerate the engine RPM increases and therefore the power increases even more (for some reason unknown to me, maybe Roger has some insight?)
Power is a function of two things-torque and RPM. By opening the throttle, you are burning more fuel/air mixture, which increases torque. You are also increasing the number of power strokes in a given period of time (by increasing RPM).
as power increases the engine needs a slightly richer mixture. So now you depart with too lean a mixture and there is what can (not will) cause that engine failure I was speaking of. This ignores the fact that the engine will also be running excessively hot. The checklists say below 3000 feet density altitude to leave the mixture rich, there is a reason for that.
It really depends on HOW lean you are running it. Like you said, the engine will (likely) be running excessively hot. However, if you have leaned far enough that you are getting reduced power (now because there is not enough fuel for all the air going into the engine), you are probably not going to have a problem with temps. You will, however have a problem getting over the trees off the departure end of the runway. The reason checklists say to leave full rich below 3000' is because that will definitely provide plenty of extra fuel to keep the engine cool. Can leaning for best power before takeoff be done below 3000'? Sure! However, I'd personally want to have a way to watch cylinder head temps during the climb (preferably probes on each cylinder, not just on one). Also, as you move to more powerful engines (especially turbocharged ones), the likelihood of overtemping a cylinder head if you don't run it rich enough on takeoff will skyrocket.
Hope this helps and hopefully Roger can verify the accuracy of this or if not let me know if I was just told this to scare the crap out of me from leaning at lower density altitudes.
I think it comes because many pilots (and many mechanics!) have a poor understanding of what is REALLY happening inside their engine with regards to mixture, power, and cooling.
"as power increases the engine needs a slightly richer mixture" You probably won't believe this so next time your in your aircraft let it run at 1000 RPM and lean for best power. At this point gradually advance your throttle and watch the degraded full power (if you can get there without the engine dieing) performance from the overly lean mixture.
If you are leaning for best takeoff power, you must do it at full throttle. Lets assume a fuel-injected engine.
There are (among various other valves, bellowses, and gadgetry) two valves inside your fuel servo. One is attached to your mixture control. When you push the mixture lever forward, this valve opens, increasing fuel flow. When you pull it all the way back, this valve closes all the way, cutting off fuel flow to the injectors. The second valve is attached to the throttle, and moves along with the throttle plate. As you push the throttle forward, this valve opens, providing more fuel to accompany the increasing airflow. Now, if the valve was purely linear, the fuel:air ratio would stay the same as the opens. However, because engine designers know that an engine needs a richer mixture at full power, the valve is not linear. In other words, if it provides 6 gph at half throttle, it might provide 14 gph at full throttle.
What does all this mean about leaning at full power? Well, because of the non-linear nature of the throttle fuel valve, if you lean for best power at 1000 RPM, you really have no idea if that is going to be best power at full throttle.
I hope this has been somewhat educational. Engine science (and the topic of aircraft systems in general) fascinates me. For further reading, consult John Deakin's columns on AvWeb