Overheating and smoke/steam coming out the vents

I wouldn't think the EGR would cause this. The EGR is used to recycle the exhaust gases back through the engine which reduces the emissions of the vehicle. Basically it reuses unburned fuel so that it doesn't get emitted to the atmosphere.

This sounds like the heater core. The heater core is what heats the air that comes into the vehicle so that you have heat. It does this by passing the air over mini radiator where the hot coolant goes through it. So when the cold air from the outside passes over this heater core, it heats the air. When the heater core leaks it allows coolant out and can cause the engine to overheat because it gets low on coolant but at the same time that hot coolant gets picked up by the air coming into the vehicle and brings it in as steam. If there is coolant in your cooling system and not just water then it should smell sweet which will confirm this is the issue.

Sounds like a heater core for sure, but also make sure that you have a good ground connection at your battery because electrolysis can be a root cause of heater core failure. You can check this by starting the vehicle once cooled off, then place a positive voltmeter lead into the coolant at the radiator or degas bottle, then the negative lead on the engine block. There should be less than 1 volt read, if there is more, repair the battery ground or add an extra ground between the battery and the chassis. I've seen repeat heater core failures due to an undiagnosed electrolysis issue.