The point here is that these sorts of "math tricks" are the next stage equivalent of what you've called "the easy parts." After a while you no longer have to stop to use the calculator, you can get into flow and get ahead with the stuff that really matters.

It's simply the next step along a long road that not everyone travels.

And by the way - I absolutely use the tools available - calculators, symbolic algebra packages, GeoGebra, and more. It's just that not having to stop as often is a huge plus.