A proof of the Goldbach conjecture would be very useful, not because the conjecture would be proven, but because of the math behind the proof. For example, it could connect two seemingly unrelated areas of mathematics. It could also lead to other discoveries (or theorems) that are of extreme interest to various people.
The key to that proof is that it won't be done with a computer. It will have to be the hard way, through human thought and perseverance, on paper. A computer might be used to write the paper, but it won't be used to solve the problem.