That is not a 'dynamical system'. What you've just described is how a basic sieve works. The correction term can be worked out (I mentioned 'inclusion-exclusion' before) and allow you to find pi(x) given say a list of primes less than sqrt(x), this has been known all the back to Legendre, sieves themselves back to Eratosthenes. This is definitely not what Snaith is refering to as mysterious.
I do have some idea how the zeros are supposed to be related to some unkown Hermitian operator (i.e. the ramndom matrix theory stuff), but I don't think I have a chance at properly explaining how this operator will be connected to a dynamical system. There's the paper by Berry and Keating "The Riemann-Zeros and Eigenvalue Asymptotics", SIAM Review, vol 41, no. 2, pp 236-266 that goes into some detail on what this dynamical system will probably look like, I've been meaning to give it a thorough read but haven't got around to it yet.