Earlier quoted context omitted.
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…
Eigendecomposition of the covariance matrix, essentially PCA, is probably the first non-trivial step in the analysis of any dataset. The idea in the comment above seems to be that it's more useful to combine some basic knowledge of statistics with simpler visualisation techniques, rather than to quickly generate thousands of shallower plots. Being able to generate thousands of plot is useful, of course, but I would a…
For a sufficiently narrow definition of "dataset", perhaps. I don't think it's the obvious step one when you want to start understanding a time series dataset, for example. (Fourier transform would be a more likely step two, after step one of actually look at some of your data.)