I think there is a lot of good feedback here. One of my first clues was the fact that the dough should be EASY to stretch / roll out. That alone illustrates the minimal gluten development, which will affect the final texture. I wonder if anyone actually uses a mixer, or if it almost has to be just very gently mixed by hand until combined and no more.
Ironically, as Jerry notes, I am getting that beautiful crusty base which exactly mimics the restaurant breads I've had. The center isn't as soft, however, and I think that I will treat this more like making a biscuit dough where overworking is absolutely detrimental to the finished product. Also possibly worth noting is that some of the US restaurant naan doesn't have as much bubbling to it.
I also wonder if I should be resting the dough longer, as in the past I haven't had a super long rest as is recommended here. I know all the restaurants I have ever seen that they have the dough on big sheet pans in the chiller, most likely having been made much earlier in the day or even day previous.
I may also try with and without the egg, but follow this recipe aside from half of the dough being without egg. It may simply be the fact that the restaurant where I detected a strong egg flavor used a lot more of it, to the point where it became imbalanced and overpowering. Worth noting is one recipe I came across actually did list it as optional, so there is that fact that it may be possible to simply omit it and have the dough still work, though obviously be different.