Earlier quoted context omitted.
I have to agree with your point about EDA. The library is neat, but even the example of covariance matrix animation is a bit contrived. Every pixel has a covariance with every other pixel, so sliding though the rows of the covariance matrix generates as many faces on the right as there are pixels in a photograph of a face. However the pixels that strongly co-vary will produce very similar right side "face" pictures.…
Aren't you missing the entire point of exploratory data analysis? Eigenfaces are an example of what you can come up with as the end product of your data exploration, after you've tried many ways of looking at the data and determined that eigenfaces are useful. Your whole third paragraph seems to be criticizing the core purpose of exploratory data analysis as though one should always be able to skip directly to the ne…
Yup this is a good summary of the intent, we also have to remember that the eigenfaces dataset is a very clean/toy data example. Real datasets never look this good, and just going straight to an eigendecomp or PCA isn't informative without first taking a look at things. Often you may want to do something other than an eigendecomp or PCA, get an idea of your data first and then think about what to do to it.
Edit: the point of that example was to show that visually we can judge what the covariance matrix is producing in the "image space". Sometimes a covariance matrix isn't even the right type of statistic to compute from your data and interactively looking at your data in different ways can help.