You are correct—photons do not have mass, and they are affected by gravity. The analogy here confounds things because it suggests gravity is part of the pictur

You are correct—photons do not have mass, and they are affected by gravity. The analogy here confounds things because it suggests gravity is part of the picture, when it is irrelevant to the explanation (example of another confounded analogy: https://xkcd.com/895/ https://xkcd.com/895/). Photons are affected by gravity because gravity affects space-time. Photons travel in straight lines, and the presence of mass changes what a straight line is. Figure out how this is connected to the Higgs and you've solved the biggest problem in physics.