Are there any subsets of the integers for which it's proven that every member is a sum of two primes? (restricted Goldbach conjecture)

Are there any subsets of the integers for which it's proven that every
member is a sum of two primes? (restricted Goldbach conjecture)

Are there any subsets of the even integers for which it's proven that
every member is a sum of at most (at most is implied hereafter) two
primes?
I don't imagine that this problem is much easier than the Goldbach
conjecture, but a proof of the weaker statement that every even number not
congruent to $\pm 1 \bmod 6$ is the sum of two primes $\implies$ every
practical number is the sum of two primes $\implies$ every even integer is
the sum of four primes. A more direct approach proving that all practical
numbers are the sum of two primes would of course lead to the same result.
I would be particularly interested to see any proofs or heuristics
implying that counterexamples to the Goldbach conjecture, supposing they
exist, should satisfy a particular congruence.