Quantity of light you can make up with higher gain (ISO) and higher available dynamic range also helps, as you mentioned.
Quality of light is determined entirely outside the camera, and better gear does not help much if at all. For photography, the quality matters countless times more than quantity, usually. I would argue the same is true for video/webcam.
This is what people don't get, and so hobbyists end up with $5k in gear, taking terrible photos. It's similar to when videographers don't put effort into their sound.
This makes it a question of how and what to light, not how much. The latter would be a matter of adjusting gain in-camera, in Post-Processing, or simply turning existing lighting up. But the former part is the important, and difficult, one.
If we don't have enough light in a scene, we need to raise ISO for a given camera and that can easily make scene look bad. For other camera with bigger and less noisier sensor, brighter lens there can be enough light and it can look very good. I can imagine that there are cases, where increasing amount of light will only improve picture for a cheap camera.
If the light is plentiful and bad, cheap camera can produce abysmal results with parts of the image completely washed white in one part, and very dark in other parts. Good sensor with high dynamic range will produce much better results. Of course, improving the light will improve both results.
But if you compare good camera in bad lighting and bad camera with good lighting the bad camera will almost always win unless it is really, really shitty.
It is also usually much cheaper to improve lighting. If your face is a shadow while the rest of the room is brightly lit there is only so much that the camera can do, regardless of how much you invest in it.
A better camera will be able to better compensate for bad lighting, but that doesn’t make the lighting good, and good lighting will improve the results of both. That’s why pro mage capturers (photogs and camera crews) spend a lot of time on getting the lighting as good as possible.
Most webcams including the ones in laptop can provide decent quality image if there is ample amount of good light on the subject.