Unless I'm misunderstanding your request, this is exactly how calendars work today.

Each calendar should be a type if appointment (i.e., sales calendar, support calendar, demo calendar, etc). Each calendar then has its own availability (this is in the calendar settings) so that type of appointment can only be booked at specifics times on specific days.

Then, each staff member has their own availability- they can only be booked at specific times on specific days. You attach a staff member to the calendars that match the type if appointments that staff member should take.

When a user goes to book, the system overlays those options and shows them ONLY times where:

1) That calendar is avaliable to recieve appointments (based on the calendar setting)

AND

2) There is an available staff member during that time block.

Feel free to reach out to me on FB if you need some assistance!