Here's the bit that really flips your pancake:
The average wait time for the non-fasttrack group stays the same, but the average wait time for the fasttrack group gets quicker - taken as a whole population, that means the average wait time is quicker.
Why?
Because the system is, broadly, in balance. New arrivals stack up at a similar rate to how the system processes people. Waiting is built-in on purpose..... so that slow periods don't see downtime. Within this system, there's room to fasttrack a proportion of the entries without causing problems.
Take a system of 5 arrivals every day, with 5 processed out every day, and a time in the system of 10 days. That's 5 in, 5 out, with 50 in the mechanism at all times. an average of 10 days for each individual.
Now assume one person each day is fasttracked - processed in 1 day instead of 10. That leaves the other 4 taking 10 days. You still have 5 in each day (1 fast), and 5 out (1 fast). The 'slow' group is still dealt with in 10 days, but the 'fast' group is through in 1. A quicker overall average.
It's really quite straightforward.
E:
Also
@dowie