No . I do not know who first used this locution, but it is relatively recent. It is meant to jar on the ear as a way of grabbing attention. t is not a common mannerism used in talking about someone's personal or professional deficiencies.
So the adjective is being used a noun object of 'do'. In your example, the speaker is saying that the person in question entirely lacks subtlety in what s/he says. S/he is probably completely tactless and doesn't 'do tact' either.
You can view it, in its origins, at least, as a sort of figurative usage of the verb 'do'. But it has always had this stretched potential. We might speak of somebody coming in to do the carpets, where what is meant is that s/he is a carpet cleaner. Margaret Thatcher said that she thought it would be possible to "do business" with President Gorbachev. So the feature you are asking about is no more than an extension of an already established freedom.
For some reason this usage has not managed to sneak into the Cambridge online Dictionary. But Collins does idiom (which is getting closer to your example) much more fully.
You can use do to say that you are able or unable to behave in a particular way.
'Can't you be nicer to your sister?'—'Nice? I don't do nice'. [VERB adjective]
It describes the usage as "informal", no doubt because it is hard otherwise to account for the use of an adjective as the object of a transitive verb. In fact, though, there is a precedent for it in ancient Greek and Latin, where the neuter form of adjectives could stand for nouns. English schoolboys would think of this a implying an unspoken noun which the adjective qualified. So our first attempts at translation would be: "I don't do a nice thing (or things of the neuter plural was used)" So it there is no reason why it could not be used poetically or rhetorically for emphasis or effect. I could even imagine a letter of application from an extremely confident applicant trying to make an impression in that way, despite the risk of irritating the old buffer that might be reading it.