Another thing to remember is that many of the beast tribes already had reasons to oppose the city-states. The Ascians just needed to exploit that in order to create enough strife to convince them to escalate to conflict. The Lambs were already doing that by calling Voidsent, who disrupt and nom aether anyway, which was the goal of having the beast tribes summon the primals.

Also, most, if not all, of the beast tribes had legends about the Paragons (the Ascians). When one shows up and unravels the secrets of manifesting your god, you listen. The same cannot be said for the Lambs, who were just people. If one showed up, they'd just think he was another denizen of the Void to be controlled/made deals with.

I also just don't think there was enough of them. Entire races of beastmen made up the worshipers of their deities, whereas the Lambs probably didn't have any substantial numbers. They probably lack the 'faith/prayer/collective willpower' element of summoning a primal.