It's not the maximum allowed climb power but it's not stupid to throttle back and level off for a while for the engine to cool down from the take-off run.
Yes, all flight manuals state reducing from full takeoff power to climb power but 2-3 psi is only a moderate cruise setting on 100 octane. And as far as keeping cool is concerned I find it's better to punch through to higher altitudes immediately because there's cooler air up higher. This assumes a flight long enough to warrant a high altitude cruise.
Does it really save fuel to climb at full (non-emergency) power?
Takeoff power is actually the full non-emergency power, climb power is somewhat less. Does it save fuel to climb at takeoff power? No. Does it save fuel to climb at max climb power? Yes. Lemme explain...
Specific fuel consumption (SFC) is lower under high load conditions meaning a lower fuel burn rate per horsepower produced (less lbs of fuel per hour per horsepower). Diesel generators on ships and in power stations typically run what they call 80/90 meaning 80% load, 90% speed because you get the most power out of the rate of fuel you're burning under these conditions. Basically, if you run into full load enrichment (also called a RICH RICH condition in aviation speak, page 33 of the Spitfire Accusim manual) you're using too much power and wasting fuel (SFC increases), this is the condition under takeoff power. We do, however, want to be as close to that point as possible because sitting on that edge means your engine is making as much power as possible per pound of fuel burned per hour and we need that power to get to altitude in a timely manner because doing so maximizes the time at cruise conditions. I assume that max climb power in your average flight manual is calculated by the engineers who built these engines to be on that aforementioned edge, correct me if I'm wrong.
Long story short, in the long run it takes the same average power (horsepower per hour) to get to any given altitude regardless of throttle, if you do it at low power it's less power and more time, at high power it's more power and less time. The resulting number will be the same every time, the difference is that in one scenario we are minimizing SFC by keeping the load high and the other we aren't.
I tend to climb at [2400, +1, lean] if the temperature is cool enough and engine is below 100. If it's above, I either enrich or reduce climb rate to get more airspeed. I typically use 160 mph for low-power climb as low power climb with 180+ mph can be rather time consuming. 180+ is good for +6 psi climbs. I have not measured my fuel consumption over an entire trip but I do notice that I'm using only half the gallons per hour while doing my yukkuri
It's important to realize that gallons per hour is not the only datum in total fuel consumption for a flight. For starters, at any given burn rate your true airspeed will be higher at higher altitude, meaning that for a given power output you will get more miles per gallon at 20,000 ft than at 10,000 ft because you are going faster for the same burn rate. Climbing quickly will put you into that sweet spot sooner so that you can minimize the fuel burned per mile for the duration of the flight. The break even point is when the fuel burned by climbing higher and fuel saved by cruising higher equals the mileage penalty of flying lower. This is why there's little point in hitting a high altitude for short trips, the climb consumes too much of the total trip time.
I'm eager to hear what any professional pilots here have to say on the matter of fuel consumption.