You've argued that if the the primes were basically random, we would expect Goldbach to hold, but as I understand it, we would expect to not be able to prove th

You've argued that if the the primes were basically random, we would expect Goldbach to hold, but as I understand it, we would expect to not be able to prove that it holds. The fact that we can prove the ternary Goldbach conjecture is (necessarily) at least as surprising as it being true: in this case it seems to be significantly more surprising. Is the significance of this result in fact the techniques used, rather than the result itself?