Re:Wall vs Roof Ventilation
Unless this is actually wrote in the locale code (which I doubt), it just some one's opinion or an HD/s uniformed judgment. There are conditions which make vertical exhaust (roof) impossible. ... Ask the HD to show you the Code section. This usually stops their argument.
The real problem with a horizontal exhaust, is the collection of grease (a fire hazard) within the duct. So the duct is required to sloped for the grease to drain to a collection point; and it must be accessible for cleaning. See the IBC Mechanical code.
In your situation the horizontal duct is very short, just slope it so the grease will drain back into the hood drip tray (if possible). And, the exterior vent should be easily removed for cleaning as required.