Ascians are not the reason why they summoned the Primals, they merely taught the bestmen how to do it.
It's like teaching someone how to use a gun. Sure, if they only knew how to throw a stone at you the results would be less grave if they decide to go crazy on you, but it's still their decision what to do with that knowledge. The beastmen were desperate before, it's just that the Ascians gave them the power to do horrible things with all their desperation and anger.
silentwindfr pretty much summed up the reasons for several tribes, they are well-known, but never adressed. For Sahagin and Kobolds, Limsa is their main opressor. Their land gets stolen, their children are killed, treaties are being ignored and if they are lucky enough to not get an axe to the head as soon as they show up even remotely in the area of humans, they get discriminated against in the worst ways possible. But we do NOTHING about it. We are like "oh yeah, I guess that happened...now let's slay their sole protector again!" and then wonder why they summon them again and again and again, draining more aether everytime.
What we SHOULD be doing is adressing those issues - find zones for the Sahagin to have their babies, give the Kobolds their land back and forbid people from stealing it on their own. Hire the Kobolds to help with the repairs from the catastrophe, since they are known to be master builders and therefore reduce prejudices against them and solve the problem of people not having anywhere to live at the same time. Of course, the one to make those kinds of decisions is the Admiral, not the WoL. But the city leaders aren't just quite in our debt anyway, our organisation is also the only one who is able to fight against the Primals. So a little threat of NOT removing the primal problem if they don't work on long-lasting solutions wouldn't hurt..
Of course, this is even more true for the tribes that are afraid of the Garleans. Since the Operation Archon, most of the Garlean bases were cut off from their homeland, no more provisions, weapon supplies, new troops or anything. In Northern Thanalan we learn that the main reason why they leave their base and go hunt on beastmen and humans alike, is because they are desperate and starving. Yet we leave the bases untouched, wreaking havoc all over Eorzea instead of finishing what we started and take those people in as war prisoners. Partially that might help once the new leader of Garlemald decides to go forth with any invasion plans, but mainly it removes the problem of the constant Garlean threat everywhere. Less human refugees the towns have to deal with, less scared beastmen to summon their primals on us. But nope, we kinda just let them sit there and that's it.
One thing I actually enjoyed about the HW scenario is that for the new primals, we did try to do something about the new Primals.
Ravana was summoned because the Gnath were afraid of the dragons who encroached on their territory and hunted them from the air. So we went and had a talk with the dragons, got them to agree to leave the Gnath alone and voilà - one problem solved.
Thordan made himself a Primal so there wasn't exactly much to do but defeat him, but we did help with the reformation of Ishgard which was direly needed to not birth someone like Thordan again.
We helped getting Shiva to understand why what she had been doing was wrong (and then gave her a free pass to turn into a Primal as many times as she wanted anyway..xD But let's ignore that part) and are on the best way to solve the conflict between the dragons and the Ishgardians which should bear fruit at some point so no more people feel like they have to defend the dragons with their life (unless Nidhogg destroys all our hard work next patch, but who knows :3).