The oxygen that can permeate into your beer post fermentation is enough to ruin it pretty quickly if you aren't very careful. Advanced home brewers and commercial brewers go to incredible lengths to prevent oxygen from contacting their finished beer. If the beer is to be consumed within a few weeks of packaging you can get away with some oxygen contamination. If a shelf life of months is desired then great care (counter pressure filling, capping on foam...) is required.
IMO unless introducing oxygen by splashing etc., it is relatively difficult to ruin a beer during bottling.
I was a winemaker before making beer, and winemakers take great care to avoid oxidation- topping up fermenters and keeping a very small headspace (the width of the bung only), using potassium metabisulfite as an antioxidant, racking extraordinarily carefully, etc.
Many, many times I've commented on the potential for oxidation in beer as well, but I will say that generally the beer is in the fermenter for far less time so the potential for severe oxidation is less.
When judging beer, the most common flaw I've noted (and other judges too) is oxidation. Most of the beers had signs of oxidation. In some cases, it was severe, like the typical "wet cardboard" we discuss. But mostly it was very early oxidation- a hint of astringency on the sides of the tongue, with a hint of staling, or a bit worse with actual "stale" flavors. It's a very common flaw.
Another issue with a large amount of headspace once fermentation slows is the potential for infection- if you notice all of the "is this infected?" threads, NONE of the beers were in a topped up carboy. Instead they were in a carboy, with lots of headspace, or in a bucket. That is another issue, of course, but it is one danger of leaving too much headspace once fermentation ends.
One of the things that has always bugged me in this forum is people saying things like "the co2 blanket will protect your beer" and "it's hard to actually aerate your beer enough to oxidize it". Both have some basis in truth, of course, but neither are totally true. The co2 "blanket" isn't going to hang around forever, and the Ideal Gas Law applies. And it really does not take a ton of aeration to get oxidation in beer. It may not ruin the beer, and the keg might be gone before it tastes too oxidized, but that doesn't mean it wasn't affected at all.