I'm not sure exactly what you're trying to get us to choose between, CoMC and ... Charms(?) or Astronomy.
Obviously they'll need Charms, so if you're thinking between CoMC and Astronomy, then I'd go for CoMC . Hagrid also looks after the animals, he knows how to heal them (as does Professor Grubbly-Plank), therefore some rudiments of animal care must be taught.
To be honest the full moon isn't that hard to calculate (just grab yourself a Muggle diary), and I would imagine Healers could consult someone else (there's probably a Percy-like person at the Ministry who sits and calculates all day). But some Potions and their ingredients are supposed to be brewed under certain celestial conditions. Whether a Healer would need Astronomy is questionable (It's not mentioned in the leaflet that Ron reads out), but an Apothocary would need it.Originally Posted by Olive Oil Med