I think the statement is correct, but the key point may be that a single crossing in the plane can be removed by using the extra topology of the projective plane.

A graph with crossing number 1 has a drawing where only two edges cross. On the projective plane, that crossing can potentially be replaced by routing one of the edges through the crosscap, giving a crossing-free embedding.

I was initially thinking about the Möbius-strip model too, and I think the important part is how the boundary is identified when turning the Möbius strip into the projective plane. It would be interesting to see an explicit construction for an arbitrary singlecross graph, rather than just a particular example.