WARNING: Spoilers for the end of the series ahead.
Rand founded an organized structure to gather and train male channelers and later established its hierarchy into Soldiers, Dedicated and full Asha'man. But that's pretty much all of his involvement with them apart calling them for battle (Rand didn't even come up with the name Black Tower), instead he let Mazrim Taim handle everything.
As the series progresses he becomes suspicious and is warned numerous times of Taim's activities but he never gets involved. Things get worse and worse until
by the time of the Last Battle Taim has become one of the Chosen and some of his students the new Dreadlords.
My question is: Why didn't Rand handle the Black Tower differently? Certainly, an army of male channelers is nothing to be looked over.
I understand that one of the reasons could be his madness and subsequent apathy. However, just before the Last Battle he heals and in about a month he tries to make things right and set a path for peace after the battle, but still he doesn't get involved with the Black Tower, even after admitting he had made some errors in that regard.
By the end of the series Androl Genhald says that Rand's lack of involvement with the Black Tower might in fact be beneficial to the Tower itself, as the Asha'man have to wean off the Dragon's shadow (or Light
;))and be an autonomous entity by themselves. But this is an unforeseen consequence, not a reason.
Related question: Why didn't Rand require fealty from the Ashaman? , but it isn't quite what I'm asking.