What do you think would happen if Marik was never born? What would the tombkeepers do without a blood related son?
You know, I could have sworn that I answered this question before, but I can't find it now.
Anyway, there are a few possibilities.
Situation (a): Mr. Ishtar gives in to the promise he made with his wife and reluctantly allows Rishid to take the initiation. Rishid starts to become an official member of the tribe and probably becomes pretty respected by everyone because he is a soft-spoken but firm person who is pretty intelligent and also kind and fair. His relationship with Mr. Ishtar is still shaky, but because Mrs. Ishtar never died in childbirth and Rishid follows all of the rules perfectly, he doesn't have a good reason to hurt Rishid emotionally or physically. Rishid becomes a great head of the Tombkeepers, but it's up in the air whether or not he manages to fulfill the Tombkeepers' mission, as who knows what kind of villain may have arisen for the pharaoh to obtain the God cards, or if one would have appeared at all. Best case scenario Atem gets the cards directly from Pegasus or Shadi after Duelist Kingdom and goes straight to Egypt where he meets with Rishid and is sent into his memories, basically skipping the Battle City plot. Worst case scenario, there is no way for Atem to receive the God cards because Rishid never leaves the Tombs to find him and so Atem is never able to pass on.
Situation (b): as Rishid's birthday grows nearer, Mr. Ishtar starts to get desperate for an heir and starts sleeping with other women in the tribe to get one. Rishid is treated even worse by Mr. Ishtar and may even be killed or permanently disabled as Mr. Ishtar is desperate to make sure Rishid doesn't become his heir. He finally gets an heir by another woman and does everything in his power to discredit his actual wife and Rishid so that he doesn't have to acquiesce to the promise he made to her.
And I think those are the two major possibilities, but I feel like there was another one that I discussed the last time I had this question (geez I wish i could find it :/)