If you found a distribution pattern in a hash function, it's no longer a cryptographic hash function.

There is no such thing in the real world as continuous, it is impossible to measure or quantify. Saying that this disqualifies the use of "gaussian" for a random distribution over a finite field is not a tenable position due to the nature of computation.

The expression "cryptographic hash function" itself implies apparent continuity of distribution. As soon as the discontinuity is found in the distribution its security is busted.