Does a glass-lens increase the amount of light entering the camera?

I had this discussion the other day with a friend of mine:
We saw this really cheap analog camera that was entirely out of plastic and even the lens didn't have any glass it was just a single piece of plastic at the front.

So my friend argued that there was no way that the camera could properly expose an image because the light entering the camera without a glass lense focusing the light wouldn't be enough. His point was that the lens would actually help to get more light into the camera.

My point was, that the lens' purpose is only to focus the light so that the image appears sharp on the film plate. In my opinion the image could be properly exposed but just wouldn't be sharp at all, kind of like a pinhole camera.

I'm not quite sure so, so I wanted to pose the question to you guys:
So you have two lenses that have the same aperture and focal length and use the same exposure time on both as well as the same film. The one is 'proper' glass lens, the other just is one piece of plastic (or has no element at all for that matter). Would the exposure be the same on both images?