Let $ G/K $ be an irreducible compact symmetric space. Let $ g $ be the pushforward onto $ G/K $ of the biinvariant metric on $ G $. Is $ g $ an Einstein metric on $ G/K $?
This is true for $ S^n=SO_{n+1}/SO_n $ where $ g $ the round metric is an Einstein metric.
It is also true for $ \mathbb{C}P^n=SU_{n+1}/SU_n $ where $ g $ the Fubini-Study metric is an Einstein metric.
Fleshing out the answer given by Johnny Lemmon:
Theorem 7.44 states: Let $ M=G/K $ be a $ G $-homogeneous isotropy irreducible space. Then (up to homotheties) $ M $ admits a unique $ G $-invariant Riemannian metric. This Riemannian metric is Einstein.
Example 7.45 of the same reference states that every irreducible symmetric space is isotropy irreducible.
Since the natural metric on $ G/K $ is $ G $ invariant then by Theorem 7.44 it is the unique $ G $ invariant metric on $ G/K $ up to homotheties and moreover it is Einstein.
Thus Thm 7.44 shows that the answer to the original question is "yes" and moreover the natural metric on any isotropy irreducible compact space $ G/K $ is Einstein.
See Theorem 7.44 in the book "Einstein manifolds" by Besse, for the case of isotropy irreducible spaces. In general, it seems that this is not true, but you should find Theorem 7.99 (ibid.) useful.