Why won't a medical MOS (68 series) take a promotable from a non-medical MOS?

It's because the proponent has decided that this is the standard of what they want to accept. You can request a grade waiver from the proponent and that will qualify you to request the training, assuming there is a seat. Sometimes proponents grant waivers to everyone, sometimes they don't.

Why do proponents restrict accession by grade? Because career progression. A promotable Soldier will be an NCO before they learn the technical aspect of their MOS. It's a risk the proponent may not want to take, especially when they can meet their accession goals with minimal waivers.

Any number of reasons. I would say mainly because a Promotable Spc. Could become a Sgt. during training. Why is that a problem, the Newly promoted NCO could not train his or her personnel, making it challenging to lead them as well.

I would think it has to do with when/if promotion orders roll in prior to graduating from the AIT. If memory serves, I think there are a large number of MOSs that don't like to take Soldiers that are Promotable.

SFC (Join to see), could you explain it better than I?