### Hole in proof presented

Yes, the process must stop eventually, but you haven't proven that the result is connected. Consider, for example, the network formed thus: take two hexagons, sharing one point. Add a point in the centre of each. Connect it to the six vertices of its hexagon. Add a line joining two vertices adjacent to the originally-shared point. This network cannot admit Step 3, but if you step-2 the last-added line, it still can't step-3, and if you next remove one of the original hexagon sides adjacent to the originally-shared point, you have a network with a cutpoint which you can disconnect with an instance of Step 3, and the invariant is then no longer maintained, because the network is disconnected.

This can happen with real polyhedra, too: consider the polyhedron formed by taking two hexagonal pyramids sharing one base vertex. Fold them, bases together, hinged at the shared point, until you can add five lines joining the other corresponding base vertices. Flatten that with one of the quadrilateral faces as the open face, step 2 four times on suitably chosen edges, and you have the network I started with.