According to Andrew Wilson, Vice President of Hangar 13, not to be confused EA's Master of Lootboxes, stated that Mafia 3 had a cold-open intro that was completely scrubbed from the release due to potentially being too "shocking".
According to a report by Eurogamer, Wilson also stated that the shocking intro was never discussed before. Apparently it was a violent prologue showing Lincoln and some of his friends being ambushed by a mob.
Things soon escalated and Lincoln had to resort to, gasp, violence to get away with his life. Wilson describes the encounter as "super violent" and it eventually resulted with Lincoln killing a cop, which turned out to be the reason why he left for Vietnam. He joined the military and served his country in order to escape murder charges.
All of this is extremely shocking, as we've never before witnessed a game where anyone mows down masses of policemen and civilians alike, only to heinously do something to avoid incarceration. In the GTA series one simply repainted a car and mass murder was instantly forgiven. At least Mafia 3's intro had a proper explanation how the protagonist dodged the long reaching arm of law.
In Hangar 13's defence, Wilson did mention that the intro was added late in development, which lead to it being rushed and eventually feeling disconnected from the game and somewhat tacked on. This was enough to cut content from a game though, so Wilson's statement that the apparent reason lay behind ethics is fairly odd. Especially when we remember that the game is titled Mafia 3 and features gratuitous amounts of violence anyway.
Wilson added that the intro felt exploitative, instead of a proper storytelling device that would grip players and let them understand Lincoln's state of mind. Apparently Hangar 13 worked on the intro for about three months before the decision was made to scrap it.
With that amount of time and enough resources, the team didn't manage to create an intro that would feel natural to the game, which is likely the actual reason why it never saw the light of day. At least they could put their backs into a proper excuse. Or not mention the scrapped intro at all. The whole situation feels odd, so if you want to try and decipher it for yourself, you can check the entire interview on Eurogamer.