I assume you agree with my argument. It's reasonable to assume that block delay in PoW network has log-normal distribution too. Let's see what will happen if we add delay due to the block mining. If there is a Poisson point process, then time delay between two arrivals in it has exponential distribution. Then if we add to random variables, one with log-normal distribution and one with exponential distribution, then we will get a random variable with log-normal distribution. So in the case of PoW we have the same type of probability distribution as in PoS! So there is no any problem here. Your original argument is invalid.