It is probably impossible to get to see all or even a significant percent of teachers, and often when it falls back to a 2nd or third choice a kid may not have seen them before, so you are correct, sometimes it can come down to a gamble. Keep in mind that people often schedule sample lessons during auditions, or even after auditions in the time period between acceptances and then having to commit, so you don’t necessarily have to do it before audition time (though that is good because you can eliminate teachers, make the list smaller, by the time auditions happens). What I would suggest as a possible course is get a list of teachers, and from that list, cull out the top X, and try and get to see them for sample lessons and such, and keep the rest as reserves. it was a bit easier with my son, he applied to relatively few schools so the number of teachers involved was pretty small as well, and with a broader list it probably will be that you will see only a relatively few teachers at the top schools on his list.