There really isn’t any argument here; recirculating range hoods just don’t make any sense. The vent pulls the air from above your stove over some charcoal filled metal mesh filters, which remove only a fraction of the contaminants from the air, then blows the still-dirty air right back into your kitchen. The prolonged food smells in a home after cooking is an indicator how infective and dirty a recirculating range hood is.
A vented range hood pulls the air from above your stove and sends it outdoors via a duct system to a roof or wall vent. This safely removes things you shouldn’t be breathing, such as carbon monoxide from gas stoves, nasty gases from cooking oil, toxins from burnt food, and also removes excess moisture which harbors food smells.