It's my opinion--I'm just a hobbyist, not a scientist--that oxygen plays a huge role in the health of planted aquariums, but does not affect algae directly. We tend to think of fish being the most needy of oxygen, and while that's sometimes the case, they aren't the only living organisms that require an abundance of oxygen. Bacteria often require larger amounts of oxygen than fish do in order to perform efficiently. If the bacteria can't do their job, then everything else in the tank suffers. The result of that is algae. The nastiest cases of algae seem to appear when the biofilter is limited and weak, which lead to eutrophic conditions. A tank can start out well because a strong biofilter isn't really needed, but as the tank matures, the larger the bacterial colonies become, and the more influence they have upon the tank. Long term success depends upon an efficient biofilter.
On the other hand, too many bacteria can hurt a tank as well. If the bacterial colonies become too large, they can literally suck all of the oxygen out of the water, suffocating fish. This is generally what happens when filters get too dirty and substrates become too nasty. This is why proper tank maintenance is crucial. In higher tech tanks, you generally see people performing regular maintenance which could be indicative of lesser algae issues than from lower tech tanks (excluding 'self-imposed' algae issues such as nutrient/lighting problems). Some algaes I've attibuted to oxygen/biofilter issues are green fuzz and BGA in some cases.
