The Sard-Smale result certainly guarantees that this will be true for a generic 𝑔, but will it hold for any 𝑔?
If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $f$ so that the pair $(f,g)$ is Morse-Smale.
And yes, there's also a proof with less heavy machinery, see for instance Theorem 6.6 in the book "Lecture Notes on Morse Homology" by Banyaga and Hurtubise.