...
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.
So, in theory, the newly generated hash might be what ends up solving a block? maybe the old one is worn out... :-) Its like playing the same lottery numbers over and over again? but I just changed the lotto numbers?
Well as I said it's random.
The effect can be positive or negative - it's random.
But it does have an effect - that you can't predict.