"He has to be there, or else something bad will happen" clearly implies that Perrin's presence must be required. Rand was not at all affected by Perrin on Dragonmount. To begin with, Perrin was in T'A'R while Rand was in the real world, so that's a huge barrier.

If Perrin had actually touched Rand, or had reached out to him mentally, that would be one thing, but it seems clear that he was there as an observer so we could see something amazing happen (and so that Perrin would know to trust Rand).