I got one of those big white square freestanding fans, and when it’s a little too close to the door, not only does airflow suffer but it’s hardly audible. I actually need the “noise” to drownd out outside noises.

It still has a few ft³ behind it, but it’s not “free” air.

Personally, I’d rather try it the cheap way to see if that works, before resorting to open-hearth surgery.