6 is "special" because it is 3# (primorial, products of the first n primes) = 2*3. The first primorial is 2, and generates all even and odd numbers through 2x + (0,1). The second primorial, 6, generates all primes greater than 3 through 6x + (1,5). Disclaimer: this formula also generates composites.
The third primorial generates all primes greater than 5 with 30x + (1,7,11,13,17,19,23,29)... And so on
The high concentration of prime distances equal to six has more to do with the rarity of "new composites" eliminated by large prime seives. 997, the largest prime sieve to establish all primes under a million, only removes 1 composite under a million. As you get further from 0, the prime patterns are largely preserved. This has led to the twin prime and k-tuple conjectures, among others.