There is no such semantic distinction between these two prepositions.
You speak of a human language as if it were a programming language disregarding its history (while even for programming languages, history is important, which is why javascript or php are in the state they are currently in). Just as no-one was thinking of the troubled American past when they were using the word "master" as a name of their default git branch or of the primary database, so no Russian (at least none that I am aware of) relies on the preposition "на" to tell them that something is or is not a country. They do, however, rely on their vocabulary of linguistic collocations, developed through the years of using the language, that tells them that the word "Ukraine" wants the preposition "на".
Were you bothered by the preposition "на"? Did you feel, back in secondary school, that saying "на Украине" somehow made it into something less than a country? Can you remember the time when it started bothering you?
No. There is no need to change language. If in any given phrase one means Ukraine as a region, like Caucasus, then it is "на", if one means Ukraine as country then it is "в". With Ukraine becoming independent country most semantic contexts though from the "region context" have naturally become "country context".
The issue here is that Russians to intentionally demonstrate their unrecognition and disrespect to Ukraine use "на" (as in the "region context") when they really speak about Ukraine as a country and thus, by Russian language rules, must have used "в".
>The whole situation, in which people get annoyed about a grammatical factoid of another people's language is just... bizarre.
It isn't annoyance. It is tens (actually looks more like early hundred) of thousands people dead for the reason that that factoid demonstrate - Russian refusal to recognize Ukrainian right to exist and to exist independent of Russia.