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c.1300, "in circumference," from phrase on round. Rare before 1600. In sense of "here and there with no fixed direction" it is 1776, American English (properly about). Of time, from 1888. To have been around "gained worldly experience" is from 1927, U.S. colloquial.
The subject is not naive, but is experienced and clever: He may look innocent, but he's been around/ Having been around the block, Sylvia not only writes stories but dispenses advice/ Oliver's been around the block and won't be seduced by money ( first variant 1920s+, second variant 1990s+)