The point is to feel like you've done something cool to your car and freed it from the shackles of The Man/the OEM intake. The primary benefit of Cold Air Intakes is to the bank account of the kit manufacturer, the secondary benefit is your car making a nicer noise, if you like the sound of an aftermarket intake. Everything else is cobblers peddled by the kind of people who sell beauty products to women. At least it's not as stupid as indexing your spark plugs.
There's been a few debunkings of CAIs over the years, this is one of the more entertaining ones- https://youtu.be/xwNtjKT_1cAthis is one of the more entertaining ones All the investigations I've seen of CAIs have shown that they usually reduce power output. OEM air intakes are generally a lot better optimised than they look and there's a lot more room for making them worse than there is for making them better.
The myth behind the CAI is that it lets you suck in denser (because it's colder) air than the stock intake, which, as you correctly pointed out, will already be sucking air from a fairly cold location. Cone filters are supposedly less restrictive than OEM paper panel filters but there's probably not much difference as the OEM filter will already provide adequate airflow for the engine in its stock state of tune. No manufacturer (maybe except GM, the renowned penny-pinchers)Few manufacturers are going to throw away 2bhp just to save 50p on the air filter. Changing the adequate OEM filter for a filter that flows more air will not make an appreciable difference because it's not a limiting factor on the engine's performance.