Short answer: Students do not have to select an instructor when booking a service such as a private lesson. If they leave the instructor as "Any" (shown as "Any preference" in the app), the request goes to all instructors who are assigned to that service and available at that time — and the first one to accept gets the booking.
How "Any" works
With "Any" selected, the student sees every time slot where at least one instructor assigned to the service has availability.
Services that need to be accepted (auto-accept off): every matching available instructor receives the request. Whoever accepts first is assigned the service, the booking is confirmed, and the student is charged using the payment method they chose.
Auto-accept services: the booking is confirmed immediately with one of the available instructors for that slot — no one needs to accept it.
If the student does pick a specific instructor, the booking/request goes only to that instructor.
Who gets the request email
The email about a new booking request (before it's accepted) goes to one address: the staff member set as the service's Request Email Recipient (Manage Studio > Services > edit the service), or the studio email if none is set.
This is the same whether the student chose "Any" or a specific instructor. The request email does not go to the instructor the student picked, and staff email notification preferences don't change this.
Once the request is accepted (or straight away for auto-accept services), the assigned instructor gets the New Service Booked email.
Emailing only the instructor the student chose isn't available yet.
Things to check
Only instructors who are assigned to the service and have availability at that time are included. If no assigned instructor is available, the slot won't appear.
You can change the instructor on a booked service afterwards from the service booking — including after it has taken place — and payroll will follow the change.