In meta-analyses, funnel-plot asymmetry tests are usually based on the association between effect size and its standard error (or variance, that in case of rank-tests are the same) or, alternatively, sample size (that is strongly associated with SE but prevents the problem of the association between the estimates of effect and of its SE). Basically, they test whether sample size is associated with the estimated effect: what we check is whether smaller sample sizes have a different (we suspect higher) estimated effect in average. However, the mechanism underlying publication bias is assumed to be the one of asymmetry: in fact, we talk about "funnel-plot asymmetry" tests. Thus, why aren't general asymmetry tests (based on the first and third moment, more rarely on other centraly tendency measures) used? At the end of the day, what we suspect is that we are missing studies that are on one tail (i.e., among the ones with lower sample size, or with higher variance, only those above/below the median value), and such one-sided lack of studies would lead to a difference between average and median and to a third central moment away from zero in the distribution of published studies, so why aren't such (in)equalities tested?
EDIT: To reply to Medwey's answer, I added a comment to clarify that I talked about the third moment and the difference between mean and median just thinking at the tests I know, but my question actually is about the lack of tests specifically focusing on tails.