In a society that enforces normative monogamy, when men fall in battle the surplus women are, yes, left nonbreeding as old maids or widows unable to remarry. Same happens when men are sworn to celibacy as monks or priests. The old maids and widows may or may not affirm their intentions by giving vows as nuns themselves.In the societies where polyandry is legal, what happens to the surplus women? Wouldn't the society suffer more from a reproductive standpoint by having a lot of virginal women than men?
Polyandry might catch on in the upper classes, though from a man's perspective, he loses far more reproductively with polyandry than a woman does with polygyny.
But a man loses even more by celibacy than he loses by polyandry.