I don’t really think that it’s white noise, it’s mostly thermal noise and “reading the sensor” noise (forgot the real name) that have very different shape from white noise.
If they were so easily filtered we’d have far superior denoise effectiveness during raw processing.
Additive white gaussian noise is quite well studied and pretty easy to remove with processing. So even if those last few bits are "just noise", if that noise is well-behaved, on the whole image you still have more information.
There's also the effect of pixel well depth. I forget the exact interplay, but I believe this usually ends up being the limiting factor before ADC resolution.
You can remove white noise with oversampling, but you can't usually oversample unless you're shooting a still-life scene and can take multiple exposures.
Otherwise, you can't strongly delineate between actual scene artifacts and noise.
Regarding pixel well depth, I suppose we can have a sensor with greater pixel size and low pixel count, like a 3840 x 2160 sensor in a full-frame camcorder. Since this would be sold as a video-oriented camera, lack of higher resolution will be a problem problem than in a still-oriented camera.