Magic in Heroes of Might and Magic V.

well, firstly, they could have simply not given them magic – nothing terrible would have happened.

As we can see in the Tribes of the East, it wasn't possible to create a race without spellcasting creatures.
And warcries are not really magic; they cannot add attack/defense (+1 from a single warcry doesn't count), they cannot curse an enemy; there is only one warcry that deals direct damage – and that's about it. The Warchief's Word does not increase initiative, and the Terrifying Roar has several analogs among creatures that are not spellcasters, meaning the effect is more physical than magical. Most warcries are aimed at increasing rage, which is a racial skill.