Pecan Sandies-Baking powder in the recipe vs. recipe without a leavening agent?
If you take the pecans out of pecan sandies, what you have left I believe is basically a short bread cookie, although those are traditionally presented in wedges.

Starting by searching for shortbread cookie recipe will give you a much better starting point.

These recipes tend to have a basis as simple as you list, although some embellish by dipping in chocolate and so on.   Others substitute cornstarch for some of the flour, to help promote the frangible, delicate texture (cornstarch has no gluten precursors like flour, so it doesn't bring risk of creating a tough or rubbery cookie).

The recipes you are reading which contain baking powder probably (without reading the recipe, its difficult to say for sure) use it to... well... literally create a small amount of leavening, or airiness in the cookie. This will contribute to its crumbly texture, especially if some toughness from overworking the flour is present.  As you have noted, it is not necessary in this type of recipe.

Side note:  cream of tartar is an acid, not a leavening agent.