I'm a little surprised at Nassim Taleb's position on this.
He has rightly pointed out that not all distributions that we encounter are Gaussian, and that the outliers (the 'black swans') can be more common than we expect. But moving to a mean absolute deviation hides these effects even more than standard deviation; outliers are further discounted. This would mean that the null hypothesis in studies is more likely to be rejected (mean absolute deviation is typically smaller than standard deviation), and we will be finding 'correlations' everywhere.
For non-Gaussian distributions, the solution is not to discard standard deviation, but to reframe the distribution. For example, for some scale invariant distributions, one could take the standard deviation of the log of the values, which would then translate to a deviation 'index' or 'factor'.
I agree with him that standard deviation is not trustworthy if you apply it blindly. If the standard deviation of a particular distribution is not stable, I want to know about it (not hide it), and come up with a better measure of deviation for that distribution. But I think the emphasis should be on identifying the distributions being studied, rather than trying to push mean absolute deviation as a catch-all measure.
And for Gaussian distributions (which are not uncommon), standard deviation makes a lot of sense mathematically (for the reasons outlined in the parent post).