We often have theory that says $\sqrt{n}(\hat\theta_n-\theta)\stackrel{d}{\to}N(0,\sigma^2),$
and then do simulations to see how well the asymptotic approximation applies. After doing so, we often present tables of the empirical mean and standard deviation of $$\hat\theta_n.$$ This doesn’t make a lot of sense.
Knowing that $$\sqrt{n}(\hat\theta_n-\theta)\stackrel{d}{\to}N(0,\sigma^2)$$ doesn’t tell us anything about the moments of $$\hat\theta_n$$ for any finite $$n$$. Convergence in distribution does not imply convergence in mean. For example,  $$\hat\theta_n$$ could be maximum likelihood estimates in a logistic regression model. These  have no finite moments for any finite $$n$$, because they are infinite with positive probability.
However, the asymptotic result does tell us that the quantiles of $$\hat\theta_n$$, suitably scaled, should converge to those of the approximating Normal distribution. The median of $$\hat\theta_n$$ should converge to $$\theta$$; the MAD of $$\hat\theta_n$$ (after scaling by $$\sqrt{n}$$) should converge to $$\sigma$$, and the probability that $$(\hat\theta_n-1.96\times\widehat{se}[\hat\theta_n],\,\hat\theta_n+1.96\times\widehat{se}[\hat\theta_n])$$ includes $$\theta$$ should converge to 95%.
Knowing how good the asymptotic approximation is for these quantile-based statistics is usually sufficient to tell us if the approximation is useful in practice. We usually don’t care about the mean of $$\hat\theta_n$$ in any substantive way, since if we did, we’d be more worried when it didn’t have one.