I have observed in many meta-analyses, and in tutorials on the matter, that the analyzed studies are commonly weighted by the reciprocal of the standard error. This seems to have a face validity because a lower standard error suggests a better estimate of the effect in question.
However, a standard error could be low for two reasons: (1) it could be because there is a larger sample size $n$ in a study; or (2) it could be because there is a lower variance in the study sample(s). Weighting by $n$ makes sense because one should weight the study not just by the more accurate estimate but also because the study is providing more evidence. And weighting by standard error would tend to be highly correlated with weighting by $n$. But why should low sample variance also matter? It could simply be a very unrepresentative sample in a study that causes the variance to be low. In that case one should not be weighting the study heavily.
Also, when doing a fixed effects meta-analysis it seems to violate the assumptions of the analysis to weight by study standard errors since one would expect there is really only one variance and all studies deviate from that variance by some amount. Once study variances are assumed equal weighting by standard error should be weighting by $n$.
So, why is weighting by standard error so often favoured over weighting by $n$?