I think, there are a few more points, and the differences between diesel and petrol engines.
A diesel engine has no throttle and always ingests the maximum amount of air. It varies the amount of fuel injected to control power, but in general, it burns with a high excess of oxygen. Heat and excess of oxygen cause production of NOx.
Adding exhaust gas to the fresh air in the intake lowers the amount of oxygen which could react to NOx. Since it also contains some water with high heat capacity from the last combustion, the total heat capacity in the cylinder increases. This means the maximum temperature during combustion is lower, and that again reduces NOx.
The one and only reason for EGR on diesel engines is the reduction of NOx. Unfortunately, a diesel engine needs the excess of oxygen and starts to produce lots of soot if it doesn't get it. So, adding exhaust gas to the point where there is no excess doesn't work.
Simple catalysts as in petrol cars since decades can not do much of it. That soot clogs them over time, and in general, they do not work with excess of oxygen. That's why diesel catalysts are complex with SCR etc.
A petrol engine is different. It has a throttle which reduces the amount of air ingested, and the exact amount of fuel, which can be burned with the amount of oxygen, is added. Theoretically, a petrol engine produces no soot and no NOx. But in reality, the fuel is never mixed perfectly with the air, so there are leaner and richer regions in the cylinder, and the engine does produce some of that stuff. But since the amount of oxygen bound in NOx is the amount needed to burn the soot (and unburned fuel), a catalyst can do a fairly well job here.
The main reason for EGR here is to reduce fuel consumption. The trottle is opened only partially most of the time, and restricts the air flow. Though this is needed to control motor power, it also means the motor has "to breathe through a straw". And that costs fuel. With exhaust gas added, the motor can breathe more freely, and consumes less fuel.
Of course, not producing NOx is better than converting it in the catalyst, which also plays a role when adding EGR to petrol engines.
(And it is not true that fuel mixture is exactly stochiometric, petrol engines also play a little with the ratio)