Reasonable people disagree about this, or just have their own personal preferences/standards without necessarily thinking others are better or worse. Essentially, this question is about how to summarise your posterior, and the idea is to get a) a point estimate indicating some kind of reasonable estimate of the posterior, and b) another number or set of numbers indicating the spread of the posterior around that point estimate, to indicate uncertainty. There are just a lot of ways of doing this!
There are some advantages to a median, in that the value indicated by the median estimate does not change with a transformation of the posterior (e.g., a log transform, it would indicate the same value, whereas the mean might change). The same applies to intervals based on a quantile from the posterior (an equal tailed interval, which is what brms provides by default, is the 2.5% and 97.5% of the posterior, and the values indicated by these also do not change when the posterior is transformed).
However, one could also use the HDI - the highest density interval. A 95% HDI indicates the 95% most likely values in the posterior, whereas the 95% ETI just indicates the middle 95% of values - not necessarily the most likely.
Things like the MAD also just indicate the spread of the value around the median.
As you can see, this can be quite an annoying thing to decide which to use. It is not the case the one is necessarily better than the other. Personally, I am often happy to use the median, mean, or even mode (most likely part of the posterior distribution) as a point estimate, and some kind of HDI as the indicator of uncertainty. However, what I would suggest is seeing if your interpretations of what is going on actually change dramatically depending on how you summarise the posterior. If your posterior has a heavy skew, then the different ways of summarising it can make a difference. Plot a density plot of the posterior and see the shape, and how the summary points line up with one another. Often, you will be relieved to see that it doesn’t make a difference. If the shape of the posterior is very skewed or has a weird shape, this might be a case where you would want to literally show a plot of the posterior so that people can see what is going on, and not be mislead by a potentially arbitrary choice of summary.
Finally, regarding using whatever %age of the posterior, this depends on how stringent you want to be about the estimate. A 99% interval contains more of the posterior than a 95%, and more again than an 89%. You can think of it like your acceptable error rate - for a 99% HDI, you expect that it is 99% likely that the true value of the underlying parameter is within that range. So the preference here depends on how conservative you want to be.