Last semesester, my professor told me about a very slick and intuitve proof of the Hairy-Ball theorem and where the euler characteristic comes itnto play. It goes a little something like this (for$2$-manifolds):
Assume you have a no-where vanishing vector field $X_p$ on your manifold $M$ and let $φ$ be its flow. Then you can find a triangulation such that the flow is transverse to the triangulation, which means that for every point at the 1 dimensional skeleton of the triangulation you have a direction where $φ$ pushes infinitesimaly the skeleton.
Now assume that you put an proton at every vertex, an electron at every edge and a neutron at every face. Then the total charge will be $χ(Μ)$.
But if you "let the wind blow" and let the flow move the triangulation, we get that now in every face we have a proton, an electron and a neutron, therefore the total charge is zero, which is absurd. Therefore, there exists a point where the field vanishes.
Now, he mentioned that Bill Thurston gave that argument. Unfortunately, our professor is a little hard to catch and ask him. I would be very interested to know if this is written somewhere. A simillar argument has been given here by Thurston himself.
See Proposition 1.3.3 in Thurston's Three-dimensional Geometry and Topology.