That's how it works in some other countries. Tevinter and Nevarra mainly. But in the rest of the world mages are considered too dangerous to be left to their own devices. Not terribly surprising, considering all the history with Tevinter, their magisters trying to dethrone the Maker, creating the first darkspawn and all that. [She waves these seemingly important details off with a wave of her wine cup.] But that's whole other story.

Anyway! Yes, other places in Thedas, especially Fereldan where I'm from, if a child starts showing signs of magic they're taken by Templars to whatever Circle is closest to them and that's where they stay for the rest of their lives, being trained by other mages and watched over by the Templars. Highborn children lose all claim to any titles or lands their families might own.

If Flemeth had been discovered to have magic she would have been taken to a Circle without question.