I would put it this way.
If you want to understand the uncertainty in the posterior of mu (and no other source of uncertainty), generating posterior predictive samples is not a good idea because the posterior predictive samples reflect the uncertainty in the posterior of mu and a bunch of other things (and there are more “other things” in more complex models).
If you are interested in asking your (now fitted) model what data sets could have been observed (other than the one you actually observed), then generating posterior predictive samples is what you want (though you may wish to avoid summarizing those posterior predictive samples with a KDE). This is not a question about mu per se, but the answer does (partially) reflect the posterior of mu.
These are both reasonable things to want, but they are different. So it really depends what are you looking for.