Right. From a definition of sortition that I find much easier to understand than your wikipedia link, it means:
the casting or drawing of lots
The direct verb form of this would be to draw lots. A more generic (and more common phrase) in English is to randomly select.
With regards to your objection to a person picking from a list not being truly random: it isn't. And if it's not actually random, then it's not random selection. I'm afraid there isn't really a simple word/phrase that differentiates between "random" and "I call it random when it's actually pseudo-random, but I believe it's close enough to random to call it random".