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In a big picture/ideal world, we'd have large sample sizes where mean and SD will be independent hence exhibiting a 'normal' distribution, as you have suggested.
Many of us, however, do not work with large samples and subsequently rely (desperately) on data transformation to be able to apply the many parametric tests we are familiar with. I feel that normality has become an artificial goal when conducting statistical analyses - to the extent that data transformation is used indiscriminately.... We seem so focused on normality and mean-based analyses that medians, ranks and non parametric alternatives, some of which are equally robust, become ignored. Further, transforming data fundamentally alters values to make them more equidistant and we can't really be sure what that does to its integrity and to its interpretability....
Research with heterogeneous groups faces the risk of losing information about its tail ends in this 'mad rush' towards achieving normality. We need to provide ample warnings about risks associated with data transformation and promote some non-parametric methods as safe alternatives.