I don't think that is the message of bioshock infinite. Ken Levine first game that he had a lot of creative control over is System Shock 2. The game was considered novel at the time because The player is betrayed by the main quest giver in the game. The first Bioshock is a remake of System Shock 2 but the setting is in a underwater city instead of a desolate space station. Bioshock Infinite is a meta commentary of how choices don't matter in games because the ending is the same regardless of player choices. Booker Dewitt always sells Anna to pay off his gambling debts, Comstock is always infertile and builds a white supremacist, Christian, genocidal ethnostate in the sky and Elizabeth actions always ends up causing more damage than it solves. Just like how the player is fated to be a mass murderer in the game, the characters are fated to make the same choices in the story. The only way to stop the choices from happening is to remove their existence from all realities/game which is the ending of bioshock infinite.

The depiction of Columbia is a standard college liberal caricature of White Christians that are patriotic. I don't think the message at the end is that White people need to disappear as part of the subtext because the erasure is only includes Comstock, Booker Dewitt, and Elizabeth. The game is also set in 1910, which is when segregation was still active and America was still a white ethnostate. The game is anti-White but not any more egregious than any other Triple A game that always cast villains as White guys and have token POC leaders.