Saitama becomes an S-rank hero

Why shouldn't everyone have an S grade?

As you said, you must be Rank 1 to be eligible for a promotion. The current rank 1 in the A-class is the hero Sweet Mask. Sweet Mask pointed out that the reason it stays at # 1 in the A class instead of moving up to the S class is to keep unworthy heroes from getting promoted.

In addition to being rank 1, you must also pass an interview. Ultimately, the decision to move up is not in the hands of the hero, but the decision of the club. Saitama had already decided that he wanted to move up, but there was still much deliberation and discussion about whether or not to move up before he was promoted.

From the One-Punch Man Wikia:

Before the interview, several Heroes Association employees observe and discuss Saitama. They argue whether or not he is a fraud. After a brief discussion, Sweet Mask is seen as an advisor on whether or not Saitama should advance. Sweet Mask replies that they don't need him for a trivial task like promoting someone to the B class. Sweet Mask also states that they only want to have a say in the promotion of people to Class A and above, as these tend to have a strong influence on the club's image.

The fact that Sweet Mask was being consulted here, rather than the club just promoting Saitama as soon as they asked for a promotion, shows that the promotions involve more than just getting ranked 1.