KL-divergence is a measure on probability distributions. It essentially captures the information loss between ground truth distribution and predicted.
L2-norm/MSE/RMSE doesn't do well with probabilities, because of the power operations involved in the calculation of loss. Probabilities, being fractions under 1, are significantly affected by any power operations (square or root), and considering we are calculating the squares of differences of probabilities, the values that are summed are abnormally small, essentially barely learning anything as the random initialization itself starts with an abnormally small loss, almost always staying constant.
L1 norm, on the other hand, does not have any power operations, making it relatively acceptable.
Loss functions, such as Kullback-Leibler-divergence or Jensen-Shannon-Divergence, are preferred for probability distributions because of the statistical meaning they hold. KL-Divergence, as mentioned before, is a statistical measure of information loss between distributions, or, in other words, assuming $Q$ is the ground truth distribution, KL-Divergence is a measure of how much $P$ deviates from $Q$. Also, considering probability distributions, convergence is much stronger in measures of Information Loss such KL-Divergence.
More clarity on the motivation behind Kullback-Leibler can be read here.