In the books, Dumbledore uses mind magic to make Mrs Cole stop asking questions about Hogwarts. I didn't want to do that here, because my Dumbledore is someone who's “well-intentioned” and genuinely cares about other people, even though ze does things that are harmful. Ze isn't supposed to disregard the lives of others for zir own convenience. (Isn't that ironic – I have to make Dumbledore nicer for this story.)

To justify the change, one could argue that it's not the most practical way to hide the existence of magic. I could imagine something like this happening: After Dumbledore leaves, some of Mrs Cole's friends ask zem about Riddle, and ze says that Tom is going to a school called Hogwarts. They try to look up Hogwarts, and it doesn't exist, so they tell Mrs Cole about it. Their only explanation is that Dumbledore must be a fraudster trying to steal children. Mrs Cole gets upset and goes to the local newspaper, who (since wizards can't infiltrate every newspaper in Britain) prints a story about it. By the time the magical world hears about it, there's an entire town full of people who will mistrust anything by the name “Hogwarts”. That would be inconvenient.

So, instead, I think it works like this: non-magical people are encouraged to know that “Hogwarts” exists and is a very exclusive boarding school. For people like Tom, Hogwarts representatives contact their guardians years earlier (under various pretenses) so that it isn't a surprise when someone comes to take the child away to school.