Also, what about "aunt" Sakura? Any idea what happens to her here? She is still Rin's sister, after all, and I don't see Rin just forgetting about her, and nor do I see Rin not telling Shirou.
While Sakura doesn't have to be important to the story, there are a lot of things you'd be missing out on. For one thing, that will reduce the amount of characters the Emisaka family can interact with from canon. Since quite a few of the major characters are servants that disappear, you can't rely on them to build a story. Sure, you can have OCs take their place - but there's more of a connection when you see a character from canon.
Sakura also has the advantage of knowing Rin and Shirou for a long time, allowing her to have a position where she can bring up some of the more embarrassing things that neither of the two would want to talk about to their kid; some of my favorite family stories are just hearing the hijinks my parents' generation got into.
Of course, if she's not married yet, you'd probably end up with a Sakura worrying about becoming an old maid - or, worse, other characters doing the worrying for her. It'd certainly be an amusing scene, for instance, to see Taiga telling her former student that she should hurry up and get hitched. Then again, if Zouken had never been taken care of, and she fulfills her role of becoming a breeding machine for the Matous... well, she might already have a kid of her own even out of wedlock.
Maybe even Shinji's.
...it'd be hard explaining that one to their kid, that's for sure.