Hey there @User44863 - great question, and good news! This is exactly how one-on-one event types are already designed to work! As soon as someone books a time with you that time will become unavailable to all other invitees and disappear from your live booking page.
Calendly is a scheduling platform that essentially sits on top of your connected personal calendar. It checks for events marked “Busy” and interprets them as conflicts, preventing invitees from scheduling new events at those times.
When an invitee schedules with you, the event is automatically published to your personal calendar and appears on your Calendly Home page under Scheduled Events. Both you and your invitee receive confirmation by email as well, helping to streamline scheduling for appointments, meetings, and more!
I hope this helps! You might benefit from our New User Guide - check it out!