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The square root plays a role in the distribution of primes as well as in the area of a circle. The former is involved in an upper bound for the numbers to be checked performing Sieve of Eratosthenes.

IIRC the pattern still arises even if you generate pseudo-primes randomly.

Yes, if you take into account all factors that might cause diagonal streaks more likely on real spiral as opposed to random grid, you might actually find out the real cause of this strange fact.