For 'true' randomness, let the passenger (or person with a gate pass), roll a 6 sided dice selected from a set. Roll a 1, get selected for secondary. Since the PAX does the randomization, the heat is off the TSA for how they got selected, assuming non-loaded dice. It'd probably reduce some of the CP stress giving the PAX a sense of control over the situation.

If 1 in 6 isn't suitable, there's other-sided dice available, but the classic cube is most familiar with all. A pair of them with other numbers selected can sit in too.

With this system, the means of selection is 100% transparent, no SSI, and yet nobody can predict the result, so it's truly random, as opposed to being capricious and arbitrary.

This may be more (or less) poplar in LAS or ACJ, depending on how you did at the tables...