Is RJ/BS saying that a man can heal himself if he is mentally ill?

I think they are saying that if you find a way, somehow, mental illness can be lived with. But it requires not only a strong personality from , but a great upbringing and the right people around you that care.

This is a world of magic, and Rand is not dealing with a normal mental illness but one caused by magical corruption. In fact, we see what must be a magical protection for him later when Nynaeve looks at the way the taint has settled on his mind.

Like @Agitelsaid, it's magic and so can't really be compared to something in our world

If anyone is looking for a metaphor, the best comparison is that Rand is going through the stages of grief.  He fluctuates through them throughout the books until he accepts that his death is inevitable. 

The funny thing is, there was no reason for Rand’s body to die.

I think his body disappearing and him living on in another body is some Prophet Elijah/Jesus stuff.

The prophecy only said his blood on the rocks. At some point his wound could break open one more time and Nynaeve could heal him fully this time, end of story. That body swapping is something weird. 

@jsbraids I would argue this is incorrect.  According to Rand's questions with the Aelfinn:

Q: "How can I fulfill the Prophecies of the Dragon and survive?"

A: "The west and the south must be as one; the north and the east must be as one. Two cannot be one; you must stand against the two as one. If you would live, you must die."

His heart could have stopped for a second or two or something. Al la Mat or some other thing like balefire.