Kind of. You don't have to generalize to all the primes if you can show that at least one of the co-prime pairs is "small." small being less than the square of the largest prime in your list.. as long as your list is all the primes up to that prime.

If the co-prime twins are evenly distributed then this shouldn't be a problem there will be an ever increasing number of twin prime candidates less than the square as the list gets longer. But the best I can do there is show that there will be PI[(p_n-1)/(p_n-2)] / 2 less than PI[pn-1] / 2

PI being the product of all the terms. Unfortunately the numbers get so massive compared to p_n^2 that I can't really say anything conclusive even though there should be plenty. I need headspace to think about larger patterns of gaps between gaps of primes. I.e. if the twin primes are far enough apart then they need to spread out enough to have some less than p_n^2.