The EGR air will be heated but the rest of the air won't be. Enough EGR to have a large percentage of heated air and the engine won't run well. EGR is typically sooty and will dirty up everything after a short time making the engine run worse.

Drastic is what won't happen with air from the radiator. Not much heat available there and it takes a long time before heat comes available. When the factory wants a WAI they take it from the exhaust. Lots of heat is available and very soon after start up. A few models use a mini heater core in the air box.