"Let's say those balls all have a random initial velocity at a random direction but all those balls are initially positioned near one corner in the cube. Thus the balls from a position stand point start with low entropy."
In every single initial configuration the particles will be at some distance from the thermometer and will cause it to read 0 according to you reasoning (until some time passes). I guess they all correspond to the same "absolute zero macrostate".
What I still don't see is how do you calculate the entropy using Gibbs entropy formula in such a way that the entropy is lower or higher for some initial configuration depending on the position of the particles. Gibbs entropy is not a property of the microstate - it's a property of the macrostate.