Firstly, “背黑锅” already existed in 1984, according to 《语言教学与研究》magazine, 1984年第01期. Because in China, 锅 is always dark bottom, so it can also be “背锅” for simplicity.
让我来仔细进行分析 “背黑锅”:
背: put something on your back.
黑锅: dark pot, 这儿的意思是responsibility that you should not shoulder, 特别是那些不好的事情。锅, is pronounced similar to “过”, which means 过错 or mistake, fault.
So 背黑锅 means take the blame, your are the scapegoat(替罪羊), or you take the responsibility for others.
So 甩锅 should come from 背黑锅/背锅, 因为锅在这儿都是responbility的意思。”甩锅” is a natural usage if you can 背锅/carry the pot on the back, then you can 甩锅/throw away the pot on your back.
“甩”的意思是throw away, 甩锅 means you throw away responsibility and hope other people to take the blame.
e.g. 她从来不甩锅! She never shirks the responsibility.
https://www.askchinax.com/question/what-is-%e7%94%a9%e9%94%85-why-does-it-mean-shift-the-blame/