Yes, some interviews might have been given away before you showed up but the school knows it has N interview slots and will distribute them accordingly, not giving them all away to those who are complete on day #1. It is all about pacing. They aren't making November interview invites in August.
After the interview, your application including your interview performance will be reviewed and scored. That's your score. When the entire interview cycle is over, the applicants will be sorted by score and they start from the top making offers. So, you have as much chance to land at the top of the ranked post-interview list as someone who interviewed on day #1.