I use Templar as shorthand for "soldiers trained to stop magic in combat of mages". Though mages policing mages is a bad idea. The Tevinter Chantry tried that, and now the Magisters rule everything with an iron, bloody fist. Same with the kingdoms having a say: they'll use the Circles as weapons. The only way to really prevent other forces from abusing it is to have the Circles and the Templar/equivalent of is by having them fully detached from other groups. Sort of like the Grey Wardens.
And the normal chantry tried using templars and now they're swallowing red lyrium, telling the church to go ef itself and going nuts. So if we're using the "If one variation of the system didn't work" then that means we can't ever trust non mages to watch over them since it didn't work this time. That's why your argument of "Well tevinter tried it and it failed" falls flat. The templar system failed too.
Also Tevinter is mostly a problem because it has a history and culture of blood magic and slavery. Take those away and the magisters wouldn't really be any worse than the nobility that rule other countries with iron fists and say FU to the peasants. Their problem isn't that there's mages ruling it's that there's slave owner sociopaths ruling.
Also, why is the risk the kingdoms might use them in war bad? Do you think the chantry doesn't use the mages when they do exalted marches? Kingdoms aren't any more likely to use mages as weapons than the chantry so it's really a lateral step. And at least mages in a circle would all be from that country they'd be made to fight for, rather than forced to fight for a religion they may not believe in.
Circles existing with mages mostly watching themselves with whatever secular forces are part of the kingdom is the best route I say. They're no more likely to get used as weapons by the chantry and it is easier for the people watching them to be regulated. If the chantry or templars abuse them then they always have the fallback of "are you questioning GOD". A state run system is a lot easier for people to protest against and publicly denounce if it messes up.
Also having the circles be run by a system like the gray wardens doesn't make sense because for one, the gray wardens do follow all the secular laws, make treaties and agreements with the secular authorities and can get kicked out by the secular authorities. So why not just cut out the middle man and have the secular authorities run something that's going to be directly effecting their mage citizens than trust some third party?