Well, yes, of course, but he's still going to show up and affect things.
Anyway, I do have an idea, or at least part of one. Rider would be a freelance Ninja or something similar, probably in a Feudal-esque Japan. Each family would be a clan fighting for control, except that the Matou clan was wiped out by Rider a while back, under the orders of someone who had hired her. Shinji and Zouken were very painfully killed, and Sakura (who had been taken in by the Matou and was being tortured, raped etc. (like in canon) to make her compliant and turn her into a perfect spy) was kidnapped and, officially, tortured to death. In reality, though, Rider took a liking to her whilst she was planning out the attack, and took her captive with the intention of giving her a happy life, using her love of torture (and Sakura's helplessness making it easy to take her) to justify it to her employer, as well as allowing her to pretend Sakura was dead.
Sakura is now a Ninja in training (having asked Rider to train her so she can repay her kindness somewhat and do something useful), as well as being in a sexual relationship with Rider. The training includes torture resistance training, which Rider very much enjoys. Rin, meanwhile, is the head of the Tohsaka house, desperately looking for the person who killed her sister in order to enact revenge. Not sure about the others, probably Shirou would be the head of another house.