What is the reason for mode collapse in GAN as opposed to WGAN? In this article I am reading:

$D_{KL}$ gives us inifity when two distributions are disjoint. The value of $D_{JS}$ has sudden jump, not differentiable at $\theta=0$. Only Wasserstein metric provides a smooth measure, which is super helpful for a stable learning process using gradient descents.

Why is this important for a stable learning process? I have also the feeling this is also the reason for mode collapse in GANs, but I am not sure.
The Wasserstein GAN paper also talks about it obviously, but I think I am missing a point. Does it say JS does not provide a usable gradient? What exactly does that mean?
I don't have a definite answer, but only a suspicion/idea:
Looking at Figure 1 from the WGAN paper, we clear see that the JS divergence on the right is not continuous at $0$, hence not differentiable at $0$. However, the EM plot on the left is continuous also at $0$. You could now argue that we have a kink there, so it should not be differentiable there either, but they might have a different notion of differentiability, I am honestly not sure about it right now.