history time! The term originated from an unnofficial court run by one Colonel Lynch, and in fact had almost zero racial connotations, as Lynch found several african-americans not guilty in his mob courts and saved them from the gallows. This was so iconic that mob vigilantism became referred to as lynching. The problem is that in the South, racist eugenics and general racism were popular, and the racists called their hate crimes lynchings by making out that being black was a crime worthy of being lynched for.
This happened so often that the term picked up racial connotations down there, because it very rarely referred to the simple mob killing of somebody perceived to have committed a heinous crime, and instead usually referred to the horrific death of black people.
It’s been a long time since then, and in our minds it’s simply reverted to being a mob killing, so we used it in the funny Mafia game without thinking. And along comes somebody who still has a kind of familial memory of the term, who is hauntingly aware of the aftermath, and they are made rightfully uncomfortable by the casual use of the term being used agressively against people.
It’s easy to conceptualise fake internet death and laugh at it and not really take it fully seriously, but less so when the word used to describe that death describes how your grandfather died.