I think part of the hardness of the cookies is the kind fat that is used. I've made snickerdoodles from the same recipe for almost 60 years. I started with shortening which was used a lot in the 40s, 50s, and 60s.
Now I use butter which or half butter and half shortening, it keeps the cookies softer and more chewy. My recipe calls for baking soda and cream of tartar.
It is one of the very favorites of my grandsons.
It's the sugar, too. On "America's Test Kitchen," they discovered that using all white sugar made for crispier cookies, and brown sugar made them chewier.