No, not a meta-discussion of online manners, but the word itself

Well, I’ll be blowed. Here have I been assuming for almost half a century that being polite, like being civil, was a by-product of living in a city (polis, civis) as opposed to the boors, churls and clowns who didn’t. It just shows: never assume, always look it up!

I made exactly the same assumption.

apparently adopted directly from the Latin and not via French

Why “apparently”?  It couldn’t possibly have been borrowed from French, since the latter has never had a -t-.

Is any language that has a word that literally means smooth going to develop a metaphorical sense meaning cultured, well-mannered? Or were these languages being influenced by the senses of the word already at work in the older Latin?

Oh, come now.  Of course these languages were influenced by the senses of the word in Latin; every word borrowed from Latin was automatically influenced by its senses in that language and could be used in any of them without the need to appeal to semantic development in the target language.  I just checked my French etymological dictionary, which says “sens mod. sous l’influence du lat. politus,” and the same could be said of any comparable borrowing in any language which used Latin as a culture language.  As for the putative generalization, Russian гладкий [gladkii] ‘smooth’ can mean ‘fluent, facile’ or ‘sleek, well-nourished,’ but not ‘cultured, well-mannered,’ so no, I don’t think it holds.