If Goldbach's Conjecture is eventually true, is it necessarily true? We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite proof. What if this number is very large? Maybe beyond our reach? 
If one proves that it is independent that eventually every even number is the sum of two primes, can we conclude that eventually every even number is the sum of two primes? Even if we may never know the $e$ from which it is Goldbach's conjecture, all the way, into infinity?
If we know that there is an $e$ from which every even number after and including $e$ is the sum of two primes, is independent, can we conclude that $e$ and every even number beyond, is the sum of two primes?
Even if we may never know what $e$ is? 
Let $e$ be the least even integer such that $e$ is the sum of two primes and for every $k=2n$ where $n \in \mathbb{N}$ where $k$ is greater than $e$, then $k$ is also the sum of two primes. Suppose it is independent of ZFC that $e$ exists. Does it follow that $e$ exists?
 A: We can't readily conclude that.  Each $\Sigma_1$ statement of arithmetic is provable in PA and hence in ZFC, but not every $\Pi_2$ statement (which is what you would seem to need here) is.
Indeed the $\Pi_1$ statement Con(ZFC) is already not provable (under the assumption that your Eventual Twin Prime Conjecture is independent). 
A: Let $S$ be the set of even integers which are not a sum of two primes. Goldbach’s conjecture is that $S$ has no members larger than $2.$. 
And, as you say, if we somehow know that it is consistent (with some system) both that it is true and that it is false, then it is true. This since, if it is false, there is a specific counter-example which can be certified by a proof. Perhaps a proof which would take more pages than there are molecules in the known universe , but still a finite proof. 
Similarly a statement such as "$S$ has no members greater than $10^{2000000}!$" is true, if independent, for the same reason.
Your hypothetical conjecture is that $S$ is finite. If that is true, that does not force existence of a finite proof.
