They are noisier because the entrance pupils of their smaller lenses are generally smaller than on the lenses that provide the same angle of view for larger sensor formats. There are physical limitations in making large entrance pupils for smaller focal lengths, as pupils bigger than f/0.5 are impossible even theoretically.
You still get more light from a smaller focal length (at the same pupil size) . Thats why we describe the amount of light coming through the lens as a function of the focal length and pupil size, or in other words we use F-stop notation.
a 10mm aperture on an 18mm lens that gives a certain fov on APS-C lets in more light than a 10mm aperture on a 28mm lens that would produce the same fov on FF.
Thats why zoom lenses that aren't constant aperture get darker when you zoom in. The pupil isnt getting smaller, you just have less light entering the lens because you are constricting the FOV.
An F-stop is and F-stop regardless of format or focal length