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Yes, exactly! Littlefinger never had reason to dig up dirt on Jon before. Even before he went to the wall, there wasn't any value in selling out a young Jon Snow, who he probably had never talked to before, to Robert. But now that Jon is KOTN and in line ahead of Sansa, he's suddenly much more interested in "how exactly" he is related to the Stark name.
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But so what?
You don't see why that knowledge would have served him more effectively when Robert was king? Robert had a public vendetta against ALL Targaryens. LF was present when Robert ordered Dany's assasination while Ned was serving as the Hand. Can you imagine what would have happened if LF told Robert that another Targaryen was alive in his realm and being protected by his friend and advisor?
LF is an agent of chaos. If disclosing the information to Robert wasn't the best means of serving his purposes, blackmailing Ned would have been another possible avenue.