If you add more noise the entropy will increase more; if you add a lot of noise the entropy will increase even more; at some high level of entropy the original signal will disappear completely. Entropy and information are reciprocal.
Self-dither does not require linearization of quantizer. Whether the signal is self-dithered or not can not be determined by math, only by ear. So, the necessity of linearization is determined by ear, not by math.
The information you are talking about is relevant only in the context of hearing. Linearization is not the main goal/necessity, it is just a means for achieving the main goal - to reduce the annoyance of granular noise of quantization.
At this stage/level of the discussion such “general” arguments are not enough. Please, elaborate - what exact premise you mean, why is it incorrect and provide the correct one.
Here I will show how linearizion of quantizer by means of dithering increase resulting quantization error. It is easy...
Linearization of quantizer by dithering results in increase of quantization error (SQNR). We need some reason for such increase. The reason is in psychoacoustics - increased error is less audible and more pleasant for hearing.