I agree with most people in that it would be much more difficult for Lucy to go untraced in the age of cell phones, internet, etc.
However, I also think things would be different for William. He gave Spirit Free School a fake name when he applied. Now, schools do extensive background checks. They would have known that the name didn't have a corresponding social security number, etc. Additionally, if he had used his real name, the termination from Waltham High would have been on his employment history. Being terminated for failure to follow curriculum would be a major red flag (even to a school with a self-guided curriculum, I would think). He may have even had his teaching license suspended if the story were set in the modern age.