....
Well actually

(no not the maths)
When you reboot a miner it gets a new unique number that's in the coinbase sig as I mentioned in that long post.
So it does actually change your hash results

Of course it's random ... but it does change it.
They way I look at it is there are two possibilities. We get a block soon, and then we can feel validated that we did something to help. We don't get a block soon, and then it doesn't matter.