When my children were young, we baked cookies every weekend. One of their favorites was 'Pecan Sandies' without the pecans. The ingredients were: 1)butter, 2)sugar, 3)flour, 4)vanilla extract, and 5)water. This recipe created a buttery cookie that was neither dry nor hard and seemed to almost dissolve in your mouth. I loved this recipe because it was so simple and contained much less sugar than most cookies. (The kids loved them straight from the oven without any added sugar for decorating.) Recently, I wanted to make these cookies with my granddaughters but I couldn't find my old recipe. Simple fix, I'll just look it up online! What I found were very few recipes with only the 5 basic ingredients I listed above. A majority of the recipes contained a leavening agent such as cream of tartar or baking powder. Now for my question: How will the cookies made with a leavening agent differ from the well loved cookies I made years ago?
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.