Yes, this kind of experience exists, but in my experience this "self-consciousness" is actually not "consciousness", but an experience related to a stage in learning. When you become a little bit aware that you cannot do something, it might temporarily make you anxious. Then, to avoid this anxiousness, you try to use your analytical mind to control your body, and it does not work really well, so you become even more anxious, creating a vicious loop.

Analytical mind is actually not very good at micro-managing the body. You can test this by trying to do something very simple, such as lifting a spoon, but this time doing it with your analytical mind.

What you describe appears to me that in your experience your analytical mind is your consciousness, and by "avoiding becoming self-conscious" you mean you want to let your body (or some other part of you, depending on what you do) do it's thing without micro-management from the analytical mind.

If you want to learn more about this, "Inner game of tennis" expains how olympic-level athletes (in this case tennis players) need to learn to let go of the control by the analytical mind when they practice. When this happens, the end result is an experience of flow. For me, being in flow state feels very good, and I can do things that would not be possible otherwise (for example in dancing). It is something I aim to experience more in my life.

Does this make us happy? Efficient? I would say so, but I would say that setting up an aim to become happy or efficient might not give you as good results as setting up an aim to experience more flow in your life.