One or two students don't make a difference. But if a number of people lie, then that dilutes the experience of the legit students. So it's not just as simple as you make it sound.

Here's an extreme example. One boy I know went to a school from kindergarten. He was tall and expected to start on the basketball team. Suddenly,in his junior year, several great players showed up from out of nowhere. There were lots of rumors about whether they were legitimately allowed to enroll, but in the end several lifelong students were bumped from the team. The coach was happy. The principal loved the idea. The kids presumably enjoyed the new school. But there were several others who lost an opportunity.

There are downsides to open door policies like you're endorsing. Most of the time, they're not so visible. But when resources are finite, people get hurt.