Rivain is not ruled by Mages. I think we have had this argument before but I will make this point again. Seers act as community leaders. They do not rule their nation anymore than the Bannorn in Ferelden do or the Council of Heralds rule Orlais. Different countries, similar functions in government. There is absolutely no evidence to suggest otherwise.
We have.
The thing about that argument is that there is no central authority to speak of. The local authorities are in charge of their areas, and rule pretty much unchallenged save for the same token fealty I've already mentioned. The Bannorn and the Council of Heralds report to a central government that can force them into line. The Seers report to one that can't.
The Jaws of Hakkon are a single group of Avvar and are not indicative of their culture as a whole. Their numbers comprised of mostly mundanes and they led by one as well. Cherry picking and obsessing over the actions of a few bad apples in a millennia old society and saying that all members of this one group should be locked up the rest of their lives is absurd.
Not when the two times something happened it was apparently an existential threat to a large country.
Interacting with Spirits is dangerous if you do not know what you are doing. You make the point yourself, the only example you have on that front is Hakkon itself. Lets examine that for a second. Mages, an offshoot group of Mages, allied and led by mundanes, a majority mundane group, manipulate a powerful Spirit into a host in order to attack their enemies. There is no backfire, the Spells and Rituals work as intended. No system was collapsed, the Avvar as a group are not harmed by Hakkon.
A: We don't actually know how safe the Avvar system is. All we know is that they (apparently) haven't had any major problems yet, which (if true) is impressive evidence of foolproofness but not proof of it. For all we know it could be exactly as good as you say, or a time-bomb long past due to explode. It would depend on how spirits being corrupted works. ("Denying them their purpose" isn't entirely specific, and we don't know that it's the only way anyway. We don't really know that Solas knows considering that the quest with Cole indicates not even he knows the exact rules for spirits changing.)
B: Let's not skate over the fact that the Avvar were trying to collapse a system, and from all I understand were thought to have the power to do so. If abominations are my main reason to look on the idea of mage freedom with fear, they're far from my only reason.
The Avvar have existed with Free Mages for far longer than the Chantry or Circle have been around and in contrast to the latter, has yet to implode. And the greatest argument you have for them to lock up their Mages is a splinter group's semi successful militarization attempt. The Chantry also militarizes Magic against it's enemies. Are the Avvar somehow worse because their Mages weren't caged beforehand? I would argue that other Mages could greatly benefit from the teachings of the Avvar in respect to Spirits.
If the Avvar caged their mages, it might have stopped this splinter faction from doing what they did. Just saying.
The thing about the way the Avvar handle spirits is that, while Solas apparently states they've never had problems, for all we know there is potential for them to do so. A spirit gets corrupted, or a demon sneaks in while they try to replace a god. Or another bunch of clowns like the Jaws of Hakkon comes along, which is a possibility I don't know that you're really giving enough weight to.
And even if the system is utterly foolproof once set up (absent JoH clowns) there's the difficulty of setting it up in the first place. For non-Avvar to set up such a system would require them to find spirits who are willing to help. That's a prime opportunity for demons. Whether or not the system works for the Avvar, spreading it risks making things a good deal worse.
Of course the fact that Mages helped helps my argument. It illustrates that even in a historically dark time that Mages and mundanes could work together to successfully rout an undesirable element and towards stability without the former being oppressed. It illustrates that before religion got in the way that Mages and mundanes were, like the Avvar, capable of a mutually beneficial societal existence.
To some degree, the Circle was mutually beneficial. As Sera points out, a lot of the mages the Circles grab would not have been guaranteed meals, an education, or a soft place to sleep if they hadn't awakened as mages. And if you live in an alienage, Templars tend to average out as a lateral move from the local police. Yeah, all of that needs changing, but as it is now, the Circles are far from the worst mages can end up with.