I always considered the Circle to have more-or-less the same logic behind it as the “war on drugs” in the USA. By identifying a threat and making the general populous terrified of it with propaganda, you can wield almost unlimited power at their expense as long as they believe that you’re keeping them safe from the bogey-man.
The Templars round up anyone with the potential for magic and put them into a concentration camp. Each is made to undergo a test and if they fail, they’re lobotomized and made into a docile slave. Those who refuse, or run away, are killed. It’s a pogrom, plain and simple, and it exists to make people believe that they need the Chantry to protect them so that the Chantry can expand its power.
The Chantry’s power grows on two fronts through the Circle pogrom. First, the populous and the secular authorities are made to fear that there might be an abomination hiding under the bed, so the Templars are allowed to do whatever they want to provided they claim that it is necessary to round up the abominable. This also provides the justification for the Chantry to keep the Templar Order as a standing army wherever it pleases, often with the result that the populous turns to the Templars for help with purely mundane matters, leading to their assumption of secular authority (like in Kirkwall, or Lothering.)
The second way in which the Chantry’s power grows is that the Circle is a powerful weapon in its own right. If the only sanctioned use of magic is within the Circle, and the Circle is controlled by the Chantry, then the Chantry is holding the biggest gun on the battlefield. You don’t suppose that might be one of the reasons that non-sanctioned magic users are so vigorously hunted down, do you? It’s not safety, it’s arms control.
The Chantry permits the Circle to train mages in approved (useful for Chantry purposes) schools of magic, and forbids all others. Blood magic is strictly taboo because it would allow a mage to bypass the need for lyrium, the flow of which the Chantry strictly controls. Other schools of magic, like shape shifting, are prohibited because they do not serve the Chantry’s objectives. Also, a shape shifter is harder to hunt down by the Templars.
Of course, for the pogrom to remain effective for almost 1,000 years requires the occasional display to keep the ignorant masses (you should know who you are, but almost certainly don’t) fearful and compliant. That part’s easy. Treat someone like a plague carrier, cage them like an animal, strip them of any and all freedoms, and inevitably someone will resist. Send the Templars to hunt them, preferably driving them into a populated area (witnesses are useful.) Make a big show of cornering the runner, and one of two things will happen: either the Templars capture the mage to be drug away in chains, making the “sheeple” feel safe, or the mage turns in desperation to blood magic to try to escape or, as a last alternative, to take out as many Templar as possible before death’s last caress. This is pure money for the Chantry, because seeing a maleficar fight for survival is exactly the sort of thing that will keep the unwashed masses terrified—not of the Templars that caused the problem, but of the mages they believe are to blame. As these stories are spread by word of mouth and become more horrible with each retelling, the demand for even more Templars will rise in a chorus most pleasing to the Chantry’s ear.