Could a girl's first period if she's a mage make her paranoid, though? I don't know how much sex ed Thedas or the Circles have. I imagine the Circles aren't too big on it simply because mage-born children are taken away so quickly. Are they rare, though? Abstinence-only education is doomed to fail, but when has that ever stopped certain types of people from trying?
But seriously. If most Circles are remotely similar to Kinloch Hold, there are going to be priests and other people reciting the Chant of Light all the time, and probably emphasizing the parts against blood magic. Templars are everywhere. I'm not sure I'd be willing to ask people what's going on with my body if that's how I'd been raised. So regardless of whether it would be an effective source for blood magic, just thinking "Maker forgive me! I became a blood mage on accident!" has got to lend itself at least somewhat to an easier possession.