A concept of a door is such that if you want to be closed off you can be and if you don't want to be you also have that choice. Having a door to the kitchen is strictly better than having no door. Personally I'd pick a glass sliding door too.

The air quality depends on exactly what you are cooking. Close the door if your cooking involves high heat, open it otherwise. And incidentally I don't understand why the author would do testing while frying an egg. Frying an egg does not involve high heat. High heat means getting close to or past the smoke point of your oil. I would not close the door when frying an egg.