It doesn't seem like such a stretch to conceptualize that if PCA assigns a tiny weight to a variable (assume all variables have been pre de-biased to mean 0, std dev 1) then it is saying it's a feature that doesn't contribute much to the overall prediction and therefore "deselecting" it by merging it with several other variables that it's correlated with, and downweighting it relative to them.

Most successful techniques I see in deep nets take the incoming features and mux them into intermediate features which are the actual ones being learned. Feature selection and PCA are in a sense just built into the network.