You need lots of outside air to come in, and lots of room air to go out.

Consider that kitchen hoods are typically mounted within 30-36” above the cooking surface. They already have the advantage that hot air rises toward them, and run from hundreds of CFM to 1200 CFM or higher. A 6” duct is recommended for a 600 CFM fan. At 1200 CFM and above, 10” is minimum. And of course you need sufficient makeup air. Without that you’ve just created a negatively pressurized room full of still air.

You mentioned a duct supplying the room with makeup air. But again if you don’t have air entering the house you’re just negatively pressurizing the house without creating air flow.