EGR and water injection do the same thing. From what I've read, both were developed for aircraft running very high compression ratios, like 14:1. Because the fuel couldn't keep up, something had to be done to reduce the combustion temperature and detonation. Both methods do that. A side effect was lower NOx, since they didn't care about emissions at the time. Remember, this is somewhere around 1940. Jetjock was right about that.
Remember what creates NOx: high combustion temperature. It's created because high temperatures rip the N2 apart into N+ which then latches onto oxygen, 'cause everybody loves oxygen. If we reduce the temperature, there's less ripping apart of N2.
One major advantage of EGR is there's no water bottle to refill.
Everybody wants EGR or the "magical" water injection to add power, or thinks that EGR is bad. What they get you is the ability to run higher compression ratios (or boost) on lower octane fuel. If you add either to an engine without changing anything else, you will lose power. You have to increase the boost or CR at the same time. It's all one big system with everything related. If you delete the EGR without telling the computer, the system won't work as well.
Just my 0.02. YMMV, etc.
Asterix