The tutor list under Book a session on the 1-on-1 Tutoring page only ever shows tutors who match the student's track: Opportunity Class and Selective students see specialist tutors for that test, and every other student sees general tutors. Nobody chooses the track and there is nothing to switch.
Reading a tutor's card
Each card shows the tutor's photo, name, specialities and subjects, any credentials, a starting price "from" per hour, and whether they offer sessions in person, online, or both. A family who has booked before sees that tutor's card marked Book again and placed first in the list.
Starting a booking
Tapping a tutor's card opens that tutor's own booking page. From there the family chooses the mode, the length and a time before setting what the session should focus on and paying. See Booking a tutoring session.
If no times are free
Availability depends on that one tutor, not on the platform as a whole, so a fully booked tutor still shows a calendar with no open times rather than disappearing from the list. Try a different day or a different tutor who offers the same track.