Planar graph via the intersection of fat thingies?
You didn't say the fat objects had to be two-dimensional, did you? Felsner and Francis prove that it's always possible with axis-parallel cubes in 3d. But, the proof involves Schramm's generalizations of Koebe-Thurston-Andreev, so it's not exactly a simpler result. They also mention along the way that for four-connected maximal planar graphs it's possible to use parallel-sided equilateral triangles.