I think it depends on the style. A west coast IPA or a heavy stout can probably tolerate a small amount of oxygen because the bitterness or malts are so strong you won't notice a little oxidation. A lighter pale ale, lager or NEIPA will suffer with even a small amount of air because, in the first two there's little to cover the off taste and, in the latter, the hops oxidize as well and they turn all kinds of ugly colors and taste stale.
And yes, as mentioned, yeast use the oxygen during fermentation. After fermentation there's nothing in there to remove the oxygen and it's free to do bad things to your beer. That said, beer doesn't oxidize immediately so if you and your friends tear through it pretty quickly (as we do here), chances are it'll be gone before it has a chance to go really bad. If you have a 5 gallon keg and it takes a month or two kick it, you might start noticing the effects after a couple of weeks, if not sooner.