Is the Goldbach conjecture any easier if we allow primes to be negative as well? That is, every even integer is the sum or difference of two primes. The twin prime conjecture talks about the occurrence of a certain kind of prime gap, but I don't know if all even integers can be a prime gap (which would imply the Goldbach conjecture for $\Bbb Z$).
Generally all the methods we have to produce gaps give us infinitely many, so the consensus seems to be that "infinitely many" is not much, or at all, harder than the "at least one" case.
Under Cramer's model this holds with probability 1.