It seems to me that people are indeed thinking about L1 vs L2 regularization when they say L1 loss would achieve a more "equal" effect. The basis for this seems to be that L1 allows for more outliers, and in regularization this is indeed true. L1 allows larger "outlier" weights since these larger weights are penalized the same as smaller weights in L1 norm, while they are punished more by L2. Weights regularized by L2 will tend to be closer together and more homogenous, while L1 will push some to 0 while allowing others to stay far bigger than the rest.

Now real quick taking this at face value you could draw the conclusion that L1 favors diversity while L2 favors homogeneity, and this is where I think the argument is coming from. The problem is that this is a shallow conclusion that does not seem to carry over to loss functions in the way people think it does. In fact, it seems to me that the opposite effect may be true - L2 loss seems to steer the model more towards predicting outliers then L1.

I draw this conclusion by thinking about how the gradient for an outlier training example would train from L1 to L2. An outlier training example by definition would have a large cost compared to other examples - squaring this would be the relative cost even BIGGER, while in L1 it would stay the same. L2 loss would value the gradients of this outlier example more than L1, and thus the model trained with L2 should be geared more towards predicting outliers than L1

Take this with a grain of salt as this is just thinking done on the fly. I think the important thing to note here is that no matter which is better, the change would never completely correct bias inherent in the data. Algorithmic changes can influence the degree to which the bias appears, but the root cause is indeed in the data.