While they may be more likely to be primes, I'm not convinced that the effect is that large. True, possible factors are restricted, but numbers in that set of possible factors are far, far more likely to be divisors than mere chance would predict. How do these balance out? It certainly isn't obvious to me.

Let's do some heuristics. the 38'th Mersenne prime is 2**6972593-1. Naively from the Prime Number theorem you'd expect to find an average of 22.12 primes in that sequence to that point. We actually got 38. So we got about 70% more primes than we were expecting to have. The limited results that we have for (10**p-1)/9 suggest a similar advantage.

Which is significant, but not astoundingly different than the naive approximation.