Well, what happens if you build a 1D-input CNN in TensorFlow and train it the usual way? Does it work? Seems like it should.

What's even the difference between 1D inputs and 2D inputs? It's all a bunch of numbers anyway. I don't think it really matters if the pixels are arranged (as you see them) in a neat rectangle vs in a straight line. You could take a 2D matrix and enumerate it as a linear string of numbers and it would still be the same matrix, just represented differently. I don't think the CNN cares either way.

I would go as far as saying that the 1D-ness of the input is just "in your head".