I would suggest that the Goldbach Conjecture is merely a reflection of the persistently preserved balance inherent in arithmetic number theory. For example, every integer has an additive inverse; every operation has an inverse; the number line maintains symmetry about a zero; etc., etc. And of course the fact that every integer is either prime or is a product of unique prime factorization - this is also a reflection of the symmetric balance maintained throughout the number theoretic "rulebook".
Goldbach pairings also reflect a balance about a midpoint and thus the conjecture appears to me as really just a translated restatement of the governing axioms and conventions. Perhaps instead of calling numbers even and odd we might try calling them balanced or unbalanced - maybe then Goldbach might seem more readily sufficient and necessary. Of course, you can't "prove" axioms - you can only conjure up equivalent statements.