The simplest way to introduce the hairy ball theorem is to study the Euler formula for graphs on the sphere. The graphs on the sphere then naturally define a vector field. For a video about this see over here
Note that in some sense this is exactly how the proof works in terms of homology since the problem is reduced to a problem for simplicial complexes.
As a fun addendum check this page
PS maybe you should put a source for all the photos?
RE: Curious Mathematics: Hedgehogs, why cyclones exist and why 3-dimensional humans can't have nice hair