You can set your frequency response using sine wave sweeps (chirps) or pink noise. Either method works good. When using REW, I find it's a little easier to wave the mic around for a while to get a spatial average while using pink noise, compared to making multiple sweeps in multiple locations and then averaging a bunch of traces in the program. However, chirps can give you polarity and phase information which pink noise cannot, so I find chirps useful for sanity fhecking the polarity and setting the time delays.
What do you mean by "verified by Eric Stevens"?
Also, you're right that periodic pink noise is a repeating pattern. That is the "periodic" part. If you have the FFT size of the RTA set the same as the sequence length of the pink noise, and if you have the window on the RTA set to "none" or "rectangular" then to get the RTA result fully updated you only need to measure for a cycle or two of the pink noise which is around a second or two. If you are doing a spatial average to capture near your left ear, right ear, and so on ... then you'll want to ensure the "averaging" feature is set to infinite and capture a cycle or two of noise at each location. So if you're waving the mic around near your left ear, we could consider that 3-5 locations and you'll want something like 5 seconds of capturing in that area. Then another 5'ish seconds near your right ear.
The exact amount of time is not critical. The important thing is can you make the measurements reapeatable? Like if you make a "measurement" using whatever method you want, can you make a second (separate) measurement and then compare the traces and have them agree really nicely? That is when you know you're moving slow enough, measuring for long enough, and doing it "right".
Pink noise should be 0dBFS meaning the loudest momentary peaks in the noise file should be at maximum level (in the recording). You'll want to also pay attention to the crest factor which is the difference between the loudest peaks and the average energy. Aim for a crest factor of 12dB if you can, and REW should let you choose this number. If you set the crest factor to a really small number in REW then it will start sounding very distorted and likely not give you a good measurement.