Also, Reid was specifically trying to keep casualties in both armies to a minimum. We don't know what the war was like before Reid and Elria stepped onto the battlefield but after their rivalry started the war basically seemed to have become a stalemate, and no one could make them budge because they were so powerful.This is actually more realistic than one might think imagining the scenario from a modern perspective. Feudal nations were not modern countries; they did not operate on the basis of "citizenship" but alliegance. Reid was a peasant, and would be expected to be loyal to his liegelord, who would have sworn a vow of fealty to his liege lord, so on and so forth up to the king. The modern concept of a unified state whose citizens are alike in purpose and specifically national identity only really came about during Napoleon's reign. It's what made his political upheavals and conscription so effective.
Reid might have been fighting for his comrades much more than whoever sat on the throne. In that case, destroying the empire wouldn't matter as much to him as protecting the army that he fought with.
The political leaders had enough sense of self-preservation that they didn't try to order either of them to go all out. Which went out the window the moment Elria died.
I think from the context it's likely that Altein started the war and was a warmongering nation in general. As time went on Altein's army came to be more loyal to Reid- who was the one that was preventing wholesale slaughter on the battlefield. And Reid didn't give a shit about what the nobles wanted.