Head position doesn't matter? Are you kidding me?
Head position absolutely does matter. The body has an ideal posture about which it moves most mechanically efficient. In theory, that is when the head is almost always neutral and/or perfectly aligned directly over top of the spine.
With that said, when the gymnast initiates the blind turn, she must lean onto the "post" arm (arm about which she is turning) and therefore, most coaches tell the gymnast to lean their ear against the arm and to look under/into the armpit.
Most of the issues (split legs, etc.) is the result of head position. When the head is out, the chest will arch, and the hips will pike. There are no two ways about it. While I've never seen it officially documented relative to adults, I believe strongly that we still have sort of an extensor reflex such that if the head goes back (cervical hyperextension) these other actions follow (chest arch, hip pike).
There are various extensor reflexes that are well documented in motor learning in relationship to babies/infants.
Anyway, if the head is out, the chest it out, and the hips pike - well...it's pretty safe to say that the body is misaligned and completing the turn will not be very easy.
Can you still complete the turn? Why sure...but, more than likely with split legs and poor alignment. So, expect to take some serious deductions in the process and don't expect to finish it in a handstand.