"They" will say that bottle caps allow oxygen ingress via osmosis.
Personally, I doubt that in most cases osmosis would allow so much oxygen ingress that it would cause appreciable oxidation in a span of a few months at proper storage conditions. I believe most oxidation occurs due to the original oxygen present in the beer and in the fill space above it. Osmosis seems a secondary effect that should take many years to have a huge effect. Also... many (most? all?) brewers are using oxygen absorbing caps these days. If those caps don't work, then well that kind of sucks and is false advertising, eh?
Besides..... canned beer goes stale as well. Leave it sit for a couple years and I can guarantee that yes it does. And there's no osmosis happening there.
In summary, I think oxidation occurs over long periods of time regardless of whether it's bottled or packaged in some other manner.
One thing I just thought of: Bottlers who are concerned about oxidation should probably be purging every bottle with CO2 prior to fill. This is easy for keggers who have a CO2 tank. Not so easy for bottlers but it could be done, and should if you want maximum storage life, say greater than 6 months like I indicated previously.
Personally, to some extent I kind of like the taste of oxidized beer anyway. Gives it a little something. The guys in the old days hundreds of years ago agreed with this as well, purposely blending new beer with stale beer to make it taste the way they liked. They'd go 50/50 or 70/30 or whatever to give it that little something that they enjoyed. I don't mind a little oxidation, especially in a stronger beer but even in a low gravity beer it's not so off-putting that I won't drink it. So there's that too. Taste is in the eye of the beholder like anything else.