The relationship between significance level and sample size is reliant on a complex set of assumptions to say the least, and, when everything is stripped away, is perhaps best seen as a way of discovering just how difficult it will be to deblur the world. What power prescription we need.

Often (always?) these constraints are all so very much more complex than Gaussian power analysis states. You do it as a way of sketching the depth of a problem I think, not much more.

The linked paper is a pretty clear introduction of the high level problems. I think it's perhaps a little more grim than necessary, but then again that might just be my own bias.