If you multiply a white texture with a white LinearGradient, the result will stay white. In order to color something red, you need to multiply it with either a red texture or a red color parameter. The desaturated LinearGradient is greyscale/white - so you need something red 
So either use a texture in the correct color (which then gets increased intensity where the LinearGradient travels), or just mutliply it with a red color parameter.
And yeah, you can replace the LinearGradient with a texture, but then take a look at Astrotronics Panner setup, because i just have noticed thanks to his post, that the Panner speed is controlled by a Vector2, instead of just one single parameter. You may not notice it with simple gradient textures, but it currently runs diagonal, because a single parameter gets interpreted equally for U and V direction (it gets moved to the side and down at the same time), and that becomes obviouls, if you mix in a more complex texture than a simple gradient.
Modified it to use a white texture, and just added a color parameter (and renamed it Emissive Color), which i set to a red color. You can see the start of the LinearGradient turning the white texture red in the preview window in the left corner.
The sine node may show a red gradient in the preview window, but the result actually is greyscale thanks to the desaturation node. You can test that by disconnecting the base color input, and just use the emissive color for a test run, you will see a grey/white gradient wandering.