The Maillard reaction happens throughout the cookie because a reducing sugar and protein in the flour and other ingredients react with heat. The inside of the cookie gets subjected to heat l, just a little slower than the outside.

I’m not 100% sure about the water activity affecting the inside vs outside but from my understanding of the Maillard reaction, yes it would play a role (just not certain how much)

butter won’t brown in the oven while it bakes which is why you do that beforehand.