The longest I have kept store bought onion sets was for about two months in a cool place buried under lightly moist soil. Can't speak to the taste of using bulbing onions as greens because I sow bunching onions seeds for green onions.

Sets are dormant and have been stored since last season. If kept cool and dry they will last a long time.

Plants are a different story. They have been grown this season and lifted from the ground just prior to shipping. They are not dormant. They need to be planted soon.

You say you "picked up a bundle" of sets. That sounds like plants, which come in bundles, not sets, which are bagged.

Jim

A bundle will last in cool place, in moist soil.
But they will rot in a week or two if left out of the soil.