It seems that the main basis for your theory is the following question:



I see this completely differently than you. IMO, the text makes it clear that it's not Rand who has this thought, but rather the narrator. The text this is based from is the following:

He had taken to wearing the sword immediately. It felt right beneath his fingers. He had told no one, no even Min, that he had recognized the weapon. And not, oddly, from Lews Therin's memories-but from Rand's own.

This was NOT Rand's personal thought, it was the narrator commenting on Rand. We had Rand's thoughts on the matter, just a paragraph earlier. How odd, that they should find this now, and make a gift of it to me, completely unaware of what they were holding.... Rand gives no indication that LTT should have any reason at all to recognize the sword, and in fact makes no mention of him period regarding this subject.

Why does the narrator say "...oddly, from...."? Aside from being a bit of poor writing with both the character and the narrator sharing the same sentiment, it's rather simple. Rand has told us, the reader, via his own thoughts, that the sword was "centuries old". There's no reason at all to imagine that Rand, in the present having been born 20 years prior, should have any knowledge whatsoever of a "centuries old" item. Additionally, throughout much of the series we've had the "LTT influence" regarding Rand's ideas/achievements/channeling/etc. LTT has been a heavy influence into Rand's knowledge to this point in the series. Therefore, a reader would expect, logically, that it was via LTT that Rand was able to recognize this sword. The "oddly", was nothing more than the narrator acknowledging this fact and clueing the readers in on it.