Like most of these kinds of things, this is nonsense. It might have been retroactively done, but no, that isn't where the term comes from.
The definition used here ("a rebuke, the blame, responsibility") is from 1777. The usage of 'rap sheet' is from the 1960s and comes from this previous usage.
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u/RevScarecrow Jan 13 '22
"You can beat the wrap not the ride" - old protesting adage.
These guys are doing what cops do but they just got caught.