I need to think about your point - initially not sure I get your point properly, so I might be confused.
I am visualising each n as in the centre of a stretch of p_n-many numbers for each given p-n.
So n-2, n-1, n, n+1, n+2 for multiples of 5 then n-3, n-2, n-1, n, n+1, n+2, n+3 for multiples of 7 and so on.
Do the probabilities _need_ to be independent of each other in the sense you mean? Surely if we look at a range of numbers n, and test each one like this, it remains true that 3/5 of the total n in the range will be in the neighbourhood of a multiple of 5 that is not n-1 or n+1, while 3/5 . 5/7 of those n will be in the neighbourhood both of a multiple of 5 not at n-1 or n+1 and at the same time in the neighbourhood of a a multiple of 7 not at n-1 or n+1 either?
I agree it is probably at this point in the argument that I am overlooking something.