I have now found a paper by A. Skopenkov where he proves the theorem from the fact that $K_5$ and $K_{3,3}$ are the only graphs $G$ with the property

For each edge $x y$, the Graph $G - x - y$ has no $K_{2,3}$ as a minor and no vertex of degree 1.

where $G - x - y$ is the graph $G$ with the vertices $x$ and $y$ removed. This is at least a starting point for finding what makes $K_5$ and $K_{3,3}$ so special.

I have now found a paper by A. Skopenkov where he proves the theorem from the fact that $K_5$ and $K_{3,3}$ are the only graphs $G$ with the property

For each edge $x y$, the Graph $G - x - y$ does not contain a $\theta$-graph and has no vertex of degree 1.

where $G - x - y$ is the graph $G$ with the vertices $x$ and $y$ removed. This is at least a starting point for finding what makes $K_5$ and $K_{3,3}$ so special.

Edit: Corrected the statement of the characterisation $K_5$ and $K_{3,3}$.

I have now found a paper by A. Skopenkov where he proves the theorem from the fact that $K_5$ and $K_{3,3}$ are the only graphs $G$ with the property

For each edge $x y$, the Graph $G - x - y$ has no $K_{2,3}$ as a minor and no vertex of degree 1.

where $G - x - y$ is the graph $G$ with the vertices $x$ and $y$ removed.

  This is at least a starting point for finding what makes $K_5$ and $K_{3,3}$ so special.

I have now found a paper by A. Skopenkov where he proves the theorem from the fact that $K_5$ and $K_{3,3}$ are the only graphs $G$ with the property

For each edge $x y$, the Graph $G - x - y$ has no $K_{2,3}$ as a minor and no vertex of degree 1.

where $G - x - y$ is the graph $G$ with the vertices $x$ and $y$ removed. This is at least a starting point for finding what makes $K_5$ and $K_{3,3}$ so special.