In Fellowship of the Ring, Galadriel (who learned from Melian) comments that she doesn't understand what Sam means by "magic". What the Elves and Maiar have over Men isn't supernatural or divine, it's sufficiently advanced technology. The Maiar have a more fundamental understanding of and connection to the world because they were present for and part of its creation, and the Elves (at least, the Ñoldor like Galadriel) were taught by them directly.