2003 vw golf overheating engine?

simple fact - they don't work on injection systems in fact they cause the ecu to run a cold start program which enriches the fuel and wastes gas, if something as cheap and simple as an air filter increased power and gave better gas mileage don't you think the manufacturers would all fit them even as an option, also on lean burn engines the colder denser air would make the mixture even more lean leading to burnt plugs and valves, the ecu/maf/temperature sensor are all mapped together upsetting the balance would cause performance to deteriorate, the injection system squirts a measured amount of gas at each cycle so more or colder air would make no difference, unlike a carburetor which supplies gas continuously, all modern air intakes cool the gas charge and cold air intakes with oiled filters can wreck the MAF by depositing oil on the MAF element and running a incorrect fuel mixture