In my short experience with piano techniques, the most useful skill to have is HAND INDEPENDENCE.
I have been able to gain some degree of finger independence through practice, but my mind can't MANAGE BOTH HANDS MOVEMEMENTS AT THE SAME TIME.
Of course the importtance of this skill varies from piece to piece. Some pieces have some simple chord progressions on the left hand.
But others, such as the ones that got me thinking about this topic - Chopin Nocturnes and Preludes left hand movements - move up and down the keybaord pretty fast in some cases.
Yes, you can get both hands separatelyd down in a short time. The trouble is when putting hands together. Your mind, or at least mine, wanders, if you tried to quickly go back and forth thinking about the hand movements separately, you are most likely going to get lost.
It's just not the untrained mind's nature to be able to perform such difficult tasks effectively. So now, things get better with practice, which is going to be a big part of your responses, but how about some tricks you've learned to improve fast?