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.

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.