I believe that the factory engineers use egr to lower peak cylinder pressure in order to avoid making too much N.O.X. (pollution), yet residual chamber exhaust gas is going to dilute and heat the charge! One of the newer pollution control devices is cooled egr, I think that the cooler egr allows more power to be made without causing an overheat condition in the engine, I believe that egr is only operated at part throttle conditions too, but I'm not up to speed on pollution control so am just thinking aloud!
EGR reduces the amount of oxygen in the cylinder (exhaust gasses are very low in oxygen since it has mostly already been combusted), thereby reducing the amount of oxygen available to combine with the nitrogen naturally in the atmosphere. It is only used at part "throttle" as during WOT you are trying to use all the displacement available to you, whereas you don't "need" the whole engine's displacement when you are cruising along at 15% max power.
Oxygen and Nitrogen will combine simply in the presence of heat, so yes the EGR cooler helps reduce the temperature and therefore the speed of that reaction. Diesel exhaust also has more leftover oxygen than gassers do making the EGR cooler even more relevant.
On gassers EGR also helps reduce pumping losses since the EGR gasses don't have to be pulled through the throttle, reducing the "work" required to pull the piston down the bore. That doesn't apply to diesels though since we don't have a throttle plate.
All this EGR theory though only applies at part throttle - at WOT you want to get all the spent gasses the hell out of the cylinder. Partly because you need the cylinder to be full of oxygen-rich air for the next burn, but also because the exhaust gasses are HOT and the incoming cold air will cool things down. So when we are talking about high drive pressures we are talking about peak operating power, where temperatures will be the highest and the risk of damage due to high EGTs will be the most prevalent. An increase of 200° due to leftover hot exhaust can definitely make the difference!