Mat was clearly experiencing a more metaphysical event. He was being possessed! He even notes that he never really even felt sick. Whatever the AS did to him was not Healing as Nynaeve would have done.

So the question becomes, if all they wanted was power to fuel the weaves, why not bring in angreal and dig up Lelaine (aside from Siuan she was the strongest woman in the Tower at the time) and Yukiri and Saerin and Pevara ... the next strongest not invovled ... hell they didn't even max out the circle. The answer is obvious ... they prioritized trust over power, something I doubt they would have done if they felt they needed more strength.