This is the opinion of a student of Lubomyr Melnyk's "continuous music".
Subconscious, unconscious playing is very important. It is not a "stage" though, as you say, nor is it quite automated (well, it doesn't have to be).
If your hands mess up whilst trying to, I don't know, go "look Ma! No hands!" and taking your mind off the piece entirely, then this is no good. Learning to internalise aspects of the motion of the piece and hands requires good finger control to begin with; some amount of finger strength, ability to hold conflicting rhythms in either hand, ability to command a deviation in one finger whilst keeping all others on the same path,..
But automation is not a goal; not for me. It is a side effect of good technique, that your hand will naturally begin to carry out a certain chord sequence or motion without specific instruction from the brain. Then your mind can focus on higher-order things, such as perceiving the harmony and rhythm as it is being played and as it is about to be played, perceiving the touch and togetherness of the fingers, so on and so on. But you have to trust your hand to actually carry out this, say, delegated task. It must be responsible! If it can't handle whatever it is you need it to handle, you have to go back and improve the strength, the relaxation; you have to ask if you are minimising tension, if the movement feels natural or unnatural; you have to ask if you understand that part of the piece. And relaxation is very important. Tension in the hand or arm is a bright signal to the mind ("! something is wrong!") and this immediately obstructs the un or sub conscious.
Feeling your way through a piece, experiencing it as it goes, is a necessary aspect. And if you can do that, really put your mind to it, then subconscious play is easier.
Slowing down the tempo is a good first thing to try. Very often, if we have insufficient control then playing quickly exacerbates this problem and makes it impossible for the control to be learned. Once the control resides within the hand, and once the hand really understands the shape of the movement it needs to follow through, speed is an easy consequence (up to a point). Speed is a consequence, not the goal. If you strive for speed, you will not truly have it, in the most relaxed possible way. There is also such a thing as overly fast tempo, even for a player with excellent hand control. Musical pieces tend to have natural ranges of tempi, where the piece can still be musical.