With a carbureted engine, it is very difficult to run lean of peak. Given that smoothness on most carbureted engines will happen near peak, enriching slightly from there likely puts you in the red box. I would not recommend this technique.
Altitude matters when it comes to anything leaning unless you have a turbo, because as you climb there is less oxygen and you need a leaner mixture setting to compensate for that. That's why we lean relative to peak EGT most of the time. More on that below.
I would say yes to what you've said above, EXCEPT that you don't do it at full throttle, you set the throttle where you want it for cruise first. Changing throttle also affects mixture, so you set throttle first, then lean the mixture.
On all of the airplanes I've flown that had the instrumentation to measure such things, the difference in fuel burn between takeoff at full rich mixture and cruise at or lean of peak was close to 50%. That is, you're burning TWICE the fuel at takeoff as you are leaned for best economy cruise.
In addition, flight school planes spend most of their time down low. Lots of time in the pattern at no more than 1000 AGL, and lots more time practicing airwork at 1500-2500 AGL. In both cases, it's probably best to leave the mixture full rich. Otherwise, a student may get a nasty surprise when they lean early in a flight and then try to practice stall recoveries and their engine quits when they try to go to full power.
For the point-to-point flying that most people do outside of the training environment, it's no longer worthwhile to leave the mixture full rich even if you're down low. See the "When to lean" section of this article:
https://www.avweb.com/features/the-pilots-lounge-30myths-for-the-last-millennium/
Be careful with terminology here. When we say "rich of peak" and "lean of peak" we're talking about peak *EGT*, not peak power. Nothing
@David Megginson said above is incorrect (because he did say peak power), but when you are lean of peak *power* you may still be rich of peak *EGT* - And in between those peaks is not a good place to be for engine longevity.
So, when you hit peak *power* you're roughly 100º *rich* of peak EGT, and by enrichening slightly from there, you're ensuring that your leanest cylinder is rich enough to be out of the red box.
Heck, start with post 1. It's all good stuff. But yes, #15 "Manifold Pressure Sucks" is by far the one I refer people to the most often. The FAA's explanations of how engines with constant speed props work are terrible, and "Manifold Pressure Sucks" clears it right up and makes it understandable.