A question about the theorem of Kervaire and Milnor that the group $\Theta^n$ is finite for $n\neq 3$

Question

In 1963, Kervaire and Milnor proved that the group $\Theta^n$ is finite for $n\neq 3$ (https://www.math.kit.edu/iag5/lehre/semgeo2014w/media/kervaire%20milnor.pdf). Here $\Theta^n$ is the group of h-cobordism classes of homotopy $n$-spheres with connected-sum operation. I want to know a brief reason that why the proof of them doesn't work for $n=3$. I tried to read the original paper, but I found that it is hard and quite long to me. Can I get a brief explanation about why does their proof does not work for $n=3$?