Dipping the high protein dough (the milk powder adds extra protein) causes a change in the protein and sugar structures at the surface and starts cooking them. That allows for the development of that extra chewy super brown crust.
You can do the same thing with say baking soda, but it's not as effective.
The lye also produces the distinctive pretzel flavor. It can be done with baking soda, as you say, and it works pretty well if you use lots of it. Like maybe 1/2 cup in a quart. Maybe even more. This also makes a safe, fun project with kids.
Glad to see enthusiasm for this old recipe, which came to me second hand from an old German baker here in Ann Arbor more than 30 years ago. I got it from another baker who started his business making these pretzels for Michigan football games. He would make many dozens and put them on a long dowel and sell them for a dollar to the crowds (>100,000) going into the Big House on Saturday afternoons.
I used to make dozens of these for our elementary school ice cream socials back in the 80's. I would get three or four parent volunteers to help in my little wholesale bakery in my attached garage, and kids on bikes would relay them to the school three blocks away. I think we made 18 dozen one year.