I've made a ton of peanut butter cookies. I use the Cook's Illustrated recipe (behind a paywall, sorry) which is amazing. It's designed to enhance the peanut flavor by adding additional salt and using chopped peanuts in addition to extra chunky peanut butter (they specifically recommend Jif brand).
Your recipe, based on mine, is all wrong. Mine uses twice that amount of butter (two sticks, salted) and sugar (one cup each of white and light brown) for the same amount of flour and eggs. It uses a cup of extra crunchy peanut butter in addition to the butter.
I don't think you can blame the peanut butter alone for the grainy texture, though. It's likely a lack of moisture content because a half cup or even a full cup of fat really isn't a lot when it comes to 2.5 cups of flour. When you refrigerate dough, it loses moisture - fridges dry things out - so whatever small amount of moisture you had is reduced even further.
Chilling cookie dough is certainly a good way to restrict the cookie's spread but overnight is probably a bit of overkill. This article from King Arthur Flour states that as little as 30-60 minutes is more than enough time to combat spread and chill the dough.
The longer you chill cookie dough, the smaller the changes become.
Call it the law of diminishing returns. The major difference is between no chilling at all vs. chilling for 30 minutes. After that, the baked cookie continues to evolve – though very gradually.
The longer it sits, the more grainy it will get, so you need to allow for that:
That’s fresh dough, at left; three-day-old dough, at right. The longer the dough chills, the drier it becomes.
As a note, the recipe used for the test above (for chocolate chip cookies) is here. It uses 2 c flour, 1 egg, 1-1/3 c sugar (half and half white and light brown), and a full cup of fat (half butter, half shortening).