A fascinating early use of the eschewing of 'racist bones' came when the 12 African American players on the Denver Broncos boycotted the team's annual dinner-dance in 1966. 1/
https://www.newspapers.com/clip/28999313/  via @_newspapers

Mike Freeman, the president of the organization which sponsored the event, said, "There is not a racist bone in the whole Quarterback Club," making the "bone" a collective property not possessed by any of the club's members. 2/

1966 seems to have been an important year in the employment of this "not a racist bone," which seems closely tied to the backlash against the Civil Rights movement. It's the first year I find a reference to the phrase in the newspapers. 3/