The only time oxygen is beneficial is immediately after pitching so yeast have necessary resources for sterol synthesis and cell wall construction during the growth phase. Even this need can be negated. Dry yeast has all the reserves it needs, having been grown in an oxygen rich environment. And experiments are ongoing at the professional level to find other means of supplying aids to sterol synthesis, such as certain lipids (which, however, prove to have their own detrimental effects.) At any other stage, oxygen is very harmful to wort and beer. It should be excluded, as far as possible, hot side and cold side. Commercially, pure O2 is sometimes not used because, when injected inline with yeast slurry during transfer to the fermenter, it is slow to dissolve, and contact with high concentrations of O2 can kill yeast; so air may be used. But at the homebrew level, using a stone in the fermenter, pure O2 may indeed give better results. A sufficient concentration of dissolved O2 can be reached with 30-60 seconds of O2 at a flow rate low enough that bubbles do not break the surface. This is good because you don't create a lot of foam, and foam proteins only get to form once. If you have foam in the kettle, transfers, aeration, and fermentation, that's all foam that won't be on top of your glass.