I don't see how this obviates the possibility that individuals can have a private language that is in principle not understandable by others, per Wittgenstein. Consider an oenophile who has a most sensitive palate, and who can describe a wine using a whole set of adjectives that are meaningless to most and which must only vaguely represent the actual sensations being enjoyed by the expert. The expert may have a whole internal vocabulary, and due to imprecision of terms, one expert's internal vocabulary may be different from all others. You could say the same for perfumers, cheese mongers, color experts and others who have extraordinary powers of sensation, and which well might be unique to the individual. And the internal language of the individual may not be intelligible by any other because their olfactory or retinal apparatus may not be exactly alike.
I also suspect that this applies to conceptualization as well as sensation. Quite possibly Einstein's internal language was unique to him.
There was an excellent MOOC on the visual system. "Light, Spike, & Sight: The Neuroscience of Vision" - https://courses.edx.org/courses/MITx/9.01x/3T2014/course/
But the illusion here is the same: that given this natural input we begin processing before birth, that these concepts extend to completely invented terms. Most folks never look there, they never wonder why a car is called a car, and there's no downside at all. It's a pernicious concept and a wickedly-difficult thing to eventually realize.
I don't see where we disagree. The only thing I'd add is that whether you have a completely private language or not, in terms of problem-solving/goal-seeking, is not important. For non-formal, non-tech things, using common words and gestures provides the quickest way forward. Once you start creating a self-consistent system of symbols representing state and behavior, though, you might actually be better off if everybody has completely different private languages. The illusion of common understanding where there is none is more dangerous than misunderstanding. There are no red lights or sirens that go off when human communication failures happen. It's all silence. It could be no other way.