I understand your point, but I think you can't break the link between a chart and its "perceptual mechanics". Because there is no pie. There are thousands of them and you would have to have a good sample to get some basic conclusions. And of course you would have to cross that with familiarity, previous knowledge and experience, etc. Too much work.

The root of all prejudice against pie chars seems to be in the studies by Croxton in 1927 and 1932. You may want to take a look at this paper for a balanced account. The author says "In my opinion, much of the adverse criticism of the pie has come from those who have wished it to do more than it could. The pie chart is a simple information graphic whose principal purpose is to show the relationship of a part to the whole. It is, by and large, the wrong choice as an exploratory device, and it is certainly not the correct choice when the graph maker or graph reader has a complicated purpose in mind, such as displaying small changes in proportion over time". I couldn't agree more.

I am designing several charts/panels with the same dataset, ("household expenditure in European countries"). Perhaps you could take a look at the one with pie charts. It is useless (because I am asking too much). Even "star glyphs", that are similar to Chernoff faces, perform better.