I don't see a problem using airstones with low lpm aquarium pumps, but I prefer larger bubbles with a higher aeration rate to get more ppm of oxygen in the brew.I can't say I've ever heard that one before.... not saying it isn't true, just never heard that.
An airstone will still make a fine tea and is a better alternative to just stirring every few hours.
And you are correct, just stirring does not generate and maintain enough enough oxygen for many microbes to flourish. Stirring or a low aeration rate produces more of a nutritional benefit than soil biology enhancement. Growers using stirring or low aeration rates for simple nutrient extraction would be better served applying their components as top/side dressing.