well, any woman that goes to the Tower is "awakened" and trained enough to be "safe" before she is set free. It's why Joline/Teslyn/Edisina took in Bethamin and Seta (at least the reason they were willing to say out loud). This is also the reason there are so many in the Kin that are so weak, they had the ability awoken (assuming they were not Wilders) and turned out to be too weak. It even seems the AS will allow a woman to progress to Accepted despite not being strong enough to become AS ... I'd guess the bottom line for AS strength is a bit blurry, so a woman like Daigian who is clearly the weakest AS could in theory be slightly weaker than a woman who go put out of the Tower because she isn't strong enough.

As to Morgase, I agree that lack of training is at least part of her problem. But even still she barely qualifies as a Channeler ... she could barely lift a scarf with air and only for seconds at a time. Seems to me any weaker than that and the ability is not really there.