It is reasonable to believe that there is a constant $c$ such that that there exists a set of prime numbers containing at most $c \sqrt{x \log x}$ elements below $x$ such that each even number is the sum of two primes from this set. So the (lower) bound of $\sqrt{x}$ you mentioned is likely not too far away from the truth.
More specifically Granville showed (Theorem 2 in Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis, Funct. Approx. Comment. Math. Volume 37, Number 1 (2007), 159-173) that this is true assuming a strong form of the Goldbach conjecture, namely that there is a positive constant $d$ such that each even number $n \gt 2$ has more than $d n/ (\log n)^2$ representations as a sum of two primes.
Essentially this came already up before on this site see Thin subbases for the primes? where among other things Mark Lewko also mentions this result.
