The new beastmen daily quests reveal that, yes, the beastmen are tempered by their Primals. It was hinted at back in 1.0 when the sylphs mentioned that the Ixal and amalj'aa started getting more hostile not long after they had been visited by the Paragons and began to summon, stating that the Primals "darken hearts" and "poison minds", but the ARR beastmen daily quests actually directly state this, especially with the Brotherhood of Ash, who
reveal that they are virtually the only remaining amalj'aa who have not been tempered by Ifrit - hence their different coloured armour. They regard Ifrit as a fraud not deserving of their worship, and their tempered kin as fools who dishonour their ancestors by preying on the weak and defenseless - the Brotherhood instead take reverence in gaining true power through battle against worthy foes. Accordingly they're raging a bitter war of attrition against the tempered amalj'aa, which is why they invite the player to help them.
It's unclear about the other beast tribes, but considering the untempered sylphs remained green and docile while those who have been tempered turned purple and are clearly malevolent, I'm likely to accept that all the Primals temper their respective beast tribes, but each differs in the extent at to how they do this, Ifrit obviously being the one to do it the most, although I assume whoever the dragonkin's Primal is would probably be worse, given it's been tempering for years.