I've read convincing posts on icmag by a guy who had a dissolved oxygen meter saying that your average aquarium airstone hardly gets any O into the water. The bubble curtain was slightly better and the disc-shaped blue airstone was the best. However, the differences in PPM were surprisingly marginal.
I still use them though because nasty anaerobic shit grows in my rez if I don't. My reasoning is that the rootzone oxygenation in a flood and drain system comes from the flood itself - all air is expelled below the floodline during irrigation and fresh oxygen-rich air is pulled in as the floodline recedes back into the reservoir. So there's little direct effect on the plants if you're flooding with oxygen-depleted solution but there may be a strong secondary effect if you're breeding pathogens in the reservoir. Will you see these secondary effects? Tough to say, it's heavily dependent on a bunch of other factors, but as Homebrewer proved, there's at least one combination of factors where it won't make a difference.
Homebrewer, just for the record, how often do you change out your reservoir? Tap or RO? H2o2? Light-proof reservoirs? These are all things I'd imagine would make a difference in your success without airstones.