It's counterintuitive to cyclists because we don't countersteer. We lean instead, generally subconsciously. If we didn't lean, it's true that we'd have to countersteer... but nobody rides that way, making it a flawed analogy at best.
You always countersteer; you're just not doing it consciously. Because of that, you're doing it very slightly. Because you're doing it slightly, you enter into turns slowly. You're not able to take sharp corners at a decent speed.
You cannot lean without countersteering first; the lean happens because you induced a fall in the desired direction, through a tiny countersteering move. With the tiny, subconscious counter-steers, you induce only small tilts. I suspect it takes multiple small counter-steering maneuvers to "build up" a decent tilt for a sharp turn, which takes time.
Once I learned to deliberately countersteer, I then started doing it all the time. I hardly take any turn (big or small) on a bicycle without pushing forward the handle-bar on that side.
If you don't make deliberate countersteering your main steering method, so that it becomes second nature, you will not be able to count on yourself to use it in an emergency.
Maybe you countersteer. I don't. I checked by riding through a puddle, turning right, and then checking the resulting trail left by the tires. No overshoot whatsoever.
Your interpretation of the trail must have missed something. We know that all turning of a bike is via countersteering. You wouldn't be able to turn right on a bike whose steering is locked out from turning left. (See video upthread.)
I’m not good downhill mtb rider but my understanding it’s really common technique there. Even on road i do that not to take a spill in sharp-ish downhill turn going at speed