I am not so sure that it cant be taught - at least to a degree.
It is like so many things in life a big element is learning how to effectively manage your time and how to prioritise. Most of the time the plot falls apart because you become overloaded and in most cases you become overloaded because of lack of planning and or lack of methodolgy to tackle the unfolding events.
In a simple case such as an engine failure because it can take us by surprise and can cause immediate panic PPL training focuses on teaching the pilot to fall back on a rehearsed drill. Part of the drill comprises prioritising, identifying and maintaining visual contact with the proposed landing site, thereby maintaining some SA as events unfold.
In just the same way getting into the habit of rehearsing the route, the alternates, the approach and during the flight developing the habit of challenging yourself to ask where am I, what is around me, in what way, if any has my plan changed etc., goes a very long way to avoiding any nasty surprises and in so doing even the less experienced pilots will do a much better job of SA.
Yes of course it comes with experience, but you learn one hell of a lot by having the old hat sitting next to you putting to you the usual questions - so have you self briefed for the approach, what are you going to do if the active changes, suppose you dont like the look of those clouds in the distance when we get a little nearer, where you going to go, what you going to do, etc.