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?
Well you could argue that it is just a corollary of the Poincaré-Hopf theorem, but I would not consider that the easiest proof. Correspondingly there are several approaches but you don't absolutely have to introduce simplicial complexes. ^^
Lengthy image sources are in the list at the bottom 👍
Nice links, by the way. I did not know "Uncyclopedia" before, although there is a conceptually similar page called "Stupidedia" in Germany :D
I think you misunderstand me. The note part in my reply is a note. My claim is that the simplest way to explain it is using the Euler formula and then show that graphs generate vector fields. All of this can be explained visually.
You don't need Poincare-Hopf, simplices etc. but they connect nicely to the previous method.