Step 1: Simulation
To start, some basic assumptions need to be made. In cross section, a 1x6 has dimensions .75” by 5.5,” while a 1x3 has dimensions .75” by 2.5”. The weights of both are negligible compared to mine. I intended to counterbore the wood to prevent the bolts from sticking out, around 1/4”. Lastly, since my fingers would go across the entire top, I assumed there was a 160 lb weight across the top of the board.
For the simulation, I assumed there are three points of contact, the two bolts and the bottom of the board. The top bolt will be represented by a ground, since it will be under tension and shear, while the bottom bolt will be represented with a sliding joint, since it can only have a shear force. The bottom point of contact will also be a sliding joint, since it can supply a force normal to the wall, but not lateral.
The first image is the simulation using ForceEffect for the 1x3.
From this we can see that the top bolt will hold a majority of the weight, with most of the force in shear, rather than tension. Looking at the components of the top force vector, there will only be 5.9 lbs of force in tension. At the base of the board, there will be 8.3 lbs of force directly against the wall.
Now, compare this to the simulation for a 1x6, shown in the second image.
Here, the numbers differ only subtly from the 1x3 case. Now, the top bolt holds slightly more weight than before, and the bottom point of contact applies a slightly smaller force on the wall.
In addition, due to the amount of flexing the wood will experience, the bottom force will be even less than predicted in the simulation. The longer 1x6 will experience more flexing than the 1x3, so the bottom force will be minimal. As a result, it makes more sense to use the 1x6 over the 1x3.