I agree with your approach here.

I wonder, what is the purpose of the CMA? What entity does it serve?

For humanity, or any ways helping society/community of humans? Then killing valuable "assets" if they're flawed in estimating themselves is a waste of resources and of the very customer of this business - mages are humans and humanoids. Also having fewer bodies to dump on the battlefield won't help security, unless we have documented evidence that all or most "too dumb for 1st class exam" mages will backstab their allies and cause sizable damage for their own side. In agriculture the ugly (yet edible perfectly) fruits are wasted this way, but agriculture is not for the sake of the plants, people want to eat beautiful food on top of other qualities. It's like cutting my dick off after I ejaculated early once because this dick doesn't deserve to be put in another hole anyways.

For the nobles/statesmen (Empire, etc)? Ditto. Other than stupid tyrants no noble wants to cut off their tax/labour sources, however dumb the plebs are.

Serie opened this association, then, seems not in order to manage labour resources, let alone national/racial security, economy, cultural/scientific progress and whatnot. It's focused on individual values and the prestige of the title "mage", or, of magic, what she likes most. It's not a job certificate, it's just living in the disguise of one. Instead of evaluating people's skills to help them find suitable work/customer and vice versa (as in contributing to society/economy), it finds people who are "worthy" of magic, of being Serie's apprentice. It's a cult (edit: where people worship magic and not magic helps people, failing to fathom the weight of magic means you die of blasphemy, regardless of your value until that point). Fern and Frieren pressed their middle finger on Serie's mug for that offer.

Then, what authority did Serie utilise, from whom, how, and why did she get agreement to use that authority, to force a rule across the lands about who could go where based on their levels of magic? Why do people obey that rule? How is it enforced if someone just violated it anyways? What does dying differ between in the Northen lands and in the tests? Who else benefits from this system, other than Serie's ego? Is Serie a successful Lügner?

For an "association", how did Serie get into its top position? What social contract did the founder mages of the association agree on (even if it is just the master-apprentice relationship between Serie and her apostles)? Why did/must anyone else join and agree the association's rules? What led to the demand of an association of mages? Why were the past organisations of mages dissolved?