So if you run it at high power and lean it and especially if you are climbing it with a lean mixture... you could do some harm. Hence climb with rich mixture in order to prevent overheating.
You want to cool the engine? Fly faster, climb slower.
Lean of peak is cooler than peak. Rich of peak is cooler than peak. Running lean doesn't mean running hot.
Climbing with a rich mixture is appropriate sometimes, in some airplanes. Not all, and not at all times.
Most manufacturers suggest leaning above several thousand feet and at less than 75% power in order to idiot-proof the engine operation. You can't do much to hurt the engine if you're leaning with less than 75% power. Accordingly, manufacturers use this as a guideline generally in dictating when leaning is recommended.
Does this mean that leaning can't be accomplished at higher power settings and at lower altitudes? Of course not.
One should lean for density altitude.
Know your engine. If you're operating a Cessna 172 with an economizer (enrichment) valve, then you need to consider the effects of that valve in the carburetor when you lean. Especially if you reduce power. If you lean at full throttle in that airplane, for example, then when you reduce power, the mixture is leaned substantially more. You might experience a power failure, or be operating at peak...because the mixture is leaned simply by reducing power. It follows, then, that in the carbureted 172, a full power takeoff is an enrichened takeoff. One can (and often should) lean for field conditions when appropriate.
Know your aircraft, conditions, and what happens when you lean, and then act accordingly.