How does one catch an ember? With air movement. How does one blow out a candle? With air movement.
What's important here is that the air movement is back-and-forth in an acoustic wave (vs. a jet). That's important, because at the start of reaction, you're locally depleting the oxygen that will be replaced by diffusion then (once the fire is really going) convection. The acoustic wave is just taking that local air and jiggling it. Yes, at the start there may be more oxygen brought to the flame, but it also spreads out the heat from the reaction and eventually depletes the oxygen too (by eventually, I mean a few acoustic cycles).
What you're really doing is straining the fluid near the combustion zone to extinguish the local reaction (see my post above).
Have you even looked at their video? Look at it:
View: https://youtu.be/vLZ3yLeYIgk?si=PrZPIT3FzdaDN38w&t=257
Look at the rubbing alcohol spraying around due to net air movement. Earlier you can see it blowing flames and smoke around! Had they had coals instead of rubbing alcohol, what do you think would happen?
Fluid movement is not, generally, reversible. Just try sucking very hard instead of blowing when you're blowing out birthday cake's candles.
As I pointed out already, even at 20 Hz (the upper limit of what you might call "infrasound"), the wavelength is 17 meters, and at the scale in question, you need to think of it as sucking and blowing alternately and not as of waves, because far less than one wave fits in the picture.
What they show in the videos are flames involving combustible vapor, which are extinguished when the ratio of fuel to air drops below the minimum that can support combustion. Too much air is mixed with not enough rubbing alcohol vapor, and the resulting mixture is not combustible so it does not burn. That exact same mechanism is wholly ineffective with embers.
