Closed environment vs open environment in brewing

The factors determining oxygenation are yeast health vs taste of the beer. Ever tasted a starter culture, or a bottle that was only bottled a couple days ago? They will be full of acetaldehyde, the cidery/green apple flavor of oxidized beer.

It is correct that the yeast needs some oxygen for replication and general healthy-ness, but at the end of the ferment, we (the humans) want ethanol and not acetadlehyde. In the absence of oxygen the yeast (with alcohol dehydrogenase and lot of other enzymes) will convert the acetaldehyde to ethanol.

In the presence of oxygen, the reaction will reverse as the yeast try to make energy from the ethanol. Given enough time all the ethanol would be converted to energy, with acetaldehyde being the first step in the conversion.

In regards to open fermentation, it works because foam and CO2 are keeping most of the oxygen out, and the Crabtree effect: in the presence of enough sugar, anaerobic metabolism happens anyway. The important thing is that the open-fermented beer gets put in a barrel before its too late.

You've already understood the general practice. Air (really oxygen) is needed in the wort prior to pitching so the yeast can grow and replicate. After pitching, a sealed environment is desired. I think the reason its OK to aerate with air (vs, say, pure O2) is that the yeast, if healthy, will usually overwhelm anything else trying grow in the fresh wort. However, once fermentation slows, other microbes could gain a foothold.

Also, and this is the crux of the answer, once the beer matures, O2 will hasten the breakdown (oxidize) of many of the nice flavor compounds that we all enjoy and lead to further off-flavors. Again, without the yeast to clean the oxygen, this is a risk.

All that said, there are breweries that ferment in open tanks. They generally have high krausen and/or use positive pressure rooms to keeps out beasties, but not all.