But I think it makes sense that Rand at least got an even better access to LTT's memories and abilities after what happened to him at Dragonmount.

Even before this event Rand seems to become more and more knowledgeable:
"The air was thin here, and he'd had trouble breathing until he'd found a way to weave Air so that it compressed slightly around him. Like the weave that warmed him, he wasn't certain how he'd done it. He vaguely remembered Asmodean trying to teach him a similar weave, and Rand hadn't been able to get it right. Now it came naturally. Lews Therin's influence, or his own growing familiarity with the One Power?" (TGS, ch.50)

Also Rand's final happiness and the conviction that he won't hear LTT's voice again indicates to me that Rand found some insight at the end, that he is more formidable now, even without the Choedan Kal (which makes thematically sense as well).