So, in the battle between the Hero Association and the Monster Association,
Saitama defeated Garou.
Why was he only promoted to Class A?
In short, Saitama is the opposite of King: King gets credit he shouldn't get, while Saitama doesn't get credit he should. The latest chapter of the webcomic involves Flashy Flash and makes things pretty clear: Flashy Flash thinks that Garou had been "weakened" by fighting with the S-class members, and so that diminishes Saitama's accomplishment. While Flashy Flash acknowledges Saitama's speed and power as "top class", and that he's far stronger than he appears, he seems convinced that he was still reliant on Garou being "weakened".
That's been the recurring story behind Saitama's advancement within the association: people assume that he's just piggybacking on the efforts of S-class heroes, as there have often been one or more such present in his fights. Sea King had Genos and several A-class and lower heroes. Saitama even encouraged people to think of him as a vulture after this so that they'd think better of other heroes. The meteor involved Genos and Metal Knight, and Bang was also present. Genos (and I think Metal Knight, too) got a promotion within S-class over the meteor, despite having been wholly ineffective. Boros had most of the S-class involved, and no one even witnessed what Saitama did there. And Garou and the Monster Association also involves much of the S-class heroes.