Why would the probabilities of being prime be an indication of the distribution of numbers around primes? If \({N}_{x}\) has a probability of \(\frac{x}{y}\) of being a prime, that means out of y numbers x are prime around that locality where \({N}_{x}\) is. Conversely, such probabilities are also good indications of the distribution of primes, for x out of y numbers are prime, around the same area.
I am not sure what Riemann and Euler did to prime numbers, but it seems overtly complicated and a pole at s=1 is ugly, especially if you don't think it effects the numbers around it by taking limits and assigning finite values to s, the complex variable.
