The methodology actually holds true for any four spins and they don't have to be consecutive. If you had a giant database of every board spin, you could say "Spin #7142, Spin #9, Spin #357 and Spin # 14000" and look them up in the table. This probability analysis above still holds true.
The "Birthday Paradox" probability increases to 1 at a much greater factor than the PYL table does because each subsequent event has a greater probability of happening. Student #2 has to match Student #1, Student #3 can match either #1 or #2, and Student #24 then has 23 chances to match the previous students.
In the PYL experiment, each spin is an independent event with the same never-changing 1/6 probability. That means it takes a much higher number of events to reach certainty. Fun Fact: The probability of the PYL 4 whammies in a row happening will never actually become 1 (guaranteed certainty) no matter how many events that you put into the model.
Since you find it fascinating, I changed the scale of the model. Note the rapid rise of the event's probability to begin with. Contrast that with the infinitesimal increases at the bottom of the table.
/Man, I truly love doing probability.