After rewatching it, again, just the other day, I'm not sure if they were there to earn the right to pick them up. The Predators seem worried by the fact the humans have accessed the weapons vault and start running ahead, like they hadn't anticipated such an eventuality occurring.
Typically, it seems like the weapons being picked up are what then triggers the eggs. It's not meant to be a challenge for the Predators to get to the guns. It's more like them retrieving them is what initiates the facility's automated procedure, like pressing an 'on' button.
So, perhaps Predators not bringing their own along was more of a ceremonial observance. Not a rite of passage. Maybe they're just that arrogant that they didn't consider the natural curiosity of humans gaining access to their weapons storage. After all, the last time it was accessed would have been by loyal priests. Can't imagine the whalers would have known how to open it, if they even ventured down to that area.