Zohran Mamdani is doing so well for a reason: a decent part of the voter base is getting increasingly fed up by the center-right politics the Democrats have been selling. Young left-wing voters really don't like the fossils currently leading the Democratic party. If the Democrats don't start selling something better than "we aren't the Republicans", they are at risk of losing yet another generation to the next right-wing populist who claims he's going to "drain the swamp".
So no, call-out culture isn't the problem: the complete lack of left-wing values is.