I thought the power was increased more by air flow versus temperature?
Engines produce power by burning fuel and oxygen. It's generally oxygen that's the limiting factor - there's no point injecting more fuel if you don't have enough oxygen to burn it (you'll run rich, have incomplete combustion, and actually lose power). Not enough fuel is WAY worse - running lean breaks engines - this is (one reason) why, in general, factory tuned cars don't run 100% duty cycle on their injectors (they're sized with some headroom).
Easier flow (less restriction) only helps if the airflow was the restricting factor to start with.
Once you don't have flow restriction (the car can ingest as much air as it wants, as quickly as it needs), then temperature becomes a factor - cold air is denser (and thus contains more oxygen molecules for a given volume of air), and colder air helps keep combustion temperatures down as well, meaning you can run more advanced timing (this is how water/meth injection works - it cools down the intake charge, as well as increasing the effective octane of the fuel/air mix, allowing even more timing advance). Cooler air and higher octane allows better controlled combustion, which allows for timing advance, which allows for more efficient burning of fuel, ultimately all meaning more power.
Back to my original point - if you're effectively not flow restricted, for a given octane level of fuel, the only way to improve power is to allow the engine to burn more fuel and/or to burn fuel more efficiently - and the only way to do that is with lower air temps in your engine. You can work to cool air down after ingesting it (this is what intercoolers do, same with WMI as I mentioned earlier), but like your health, prevention is better than cure, and if you can start with a lower air temp at your air filters, you're going to (all other things being equal) end up with cooler air in your intake manifold after your turbos and intercooler and WMI and whatever else you try.