I guess so long as your increment is constant there are primes you’re never going to find. If a prime happens to be 2^64 from another prime you’ll still never see it with my approach.

Probably the most reasonable thing to do would be to use a Mersenne twister or other PRNG to generate a stream of random bytes. Would be plenty fast and should hopefully have no relevant pattern. No reason you should need real randomness from the OS after the initial seed.