Alright, let's sum up:

An 'old Aes Sedai' never reached her full potential. This Aes Sedai had a lot of scenes in the books. There is some sort of parallel with Elayne. Um. Elayne spends a lot of time messing around with ter'angreal, and is in possession of an angreal for much of the later books. Similarities with Cadsuane there. Maybe channeling through an angreal 'stretches' you less than normal channeling? You're using less of your own power, but it's amplified so you don't realise it?

The failing to reach her potential, was that something the Aes Sedai consciously chose or something that just happened? I can't think of any logical reason why an Aes Sedai would want to be less powerful than she could be, especially with how much the hierarchy effects everything. Unless she was already more powerful than nearly everyone, which hints at Cadsuane again. Or unless she was doing it for political reasons. Second-in-command Galina not wanting Alviarin to see her as a threat? But Galina isn't old and I doubt a Black sister would ever limit her own power.

And that bit about her teachers/superiors not pushing her like they normally would, that's just odd. Egwene, Elayne and Nynaeve had an unusual amount of leeway and freedom during their training but that was unprecedented, I can't see the same having been true for the old Aes Sedai. It's possible that she skipped the Tower as soon as she was raised, like Moiraine did, but they can't have been the only sisters in Tower history to do that.

I am very puzzled.