Is this "over-negation" a grammatical phenomenon or pure coincidence?

This is a good question. What you discribe is a common phaenomenon, at least in Indo-European languages. Consider how the words immoral and amoral have arbitrarily divided the two possible meanings of "not moral" between them. Compare also inordinate and unordered. The issue here is that it is often not unambiguously clear what comprises the negation or opposite of a concept. The prefix in- creates a word that negates or opposes the original word. But what does negation entail exactly?

Suppose you had a collection of two concepts: hot and cold. Then uncold means it must be hot.

But, if you are aware of the concepts "hot, lukewarm, neutral, cool, cold", then uncold could mean "hot, lukewarm, neutral, or cool", or it could mean "hot": it depends on your interpretation of what the available options are, and on what the opposite of cold is. If I replace uncold with not cold, then you are probably even less likely to think it must mean "hot".

Hot and cold are relatively simple concepts that have clear opposites. What if I say not men, without specifying what collection you should pick from? Does that mean "children"? Or "women"? Or "gods"?

In order to interpret "not x" or "the opposite of x", you need context, a frame of reference. Sometimes the possible interpretations are artificially divided between different suffixes, such as un-, in-, and a-. See also this question:

What types of antonyms are there?