I doubt the Taint was the origin of Rand's recycled memories, though it may well have been the cause (or partly the cause) of the LTT voice/persona.

I had a theory once, a long time ago, that the extended memories came from the barrier that guarded Callandor. The creators of the barrier must certainly have woven some recognizing part into the web which allowed the true reborn Dragon to pass but none other. My theory was that this feature included a small portion of LTT's conscience or soul or something similar to that - some sort of highly personal imprint, at least - which would release the weave when it met a perfect match. Then, as an unexpected side effect, this imprint was transferred to Rand as he claimed the sword. This shred of conscience, foreign to Rand's own mind, was not accepted into the system. Over time, possibly aided by the Taint, the imprint became the persona of LTT.

Rand's apparent communication with this second conscience might have been the imprint trying to gain acceptence into the mainframe. Or possibly, if you want to go that way, the imprint opened some sort of channel back to its source. But that I find more unlikely.

After all, it is said that Callandor is considered important in the prophecies, but the Choedan Kal is not mentioned. This could then be partly why.