Would Martin Luther King be “shouted down” today?

This in fact would be quite likely to happen. No one with a moderate view, a middle of the road and well-balanced perspective on racial matters is seen to be acceptable for most people who have been persuaded they need to take sides and anyone who differs needs to be ostracized and quite likely demeaned and diminished in their influence and effectiveness if this can be arranged. This can be accomplished through confronting them and sidelining them through sheer force brought to bear to box them in and make them cower when they are opposed too greatly.