The recent progress in image recognition which was mainly made by changing the approach from a classic feature selection – shallow learning algorithm to no feature selection – deep learning algorithm wasn’t only caused by mathematical properties of convolutional neural networks. Yes – of course their ability to capture the same information using smaller number of parameters was partially caused by their shift invariance property but the recent research has shown that this is not a key in understanding their success.
In my opinion the main reason behind this success was developing faster learning algorithms than more mathematically accurate ones and that’s why less attention is put on developing another property invariant neural nets.
Of course – rotation invariance is not skipped at all. This is partially made by data augmentation where you put the slightly changed (e.g. rotated or rescaled) image to your dataset – with the same label. As we can read in this fantastic book these two approaches (more structure vs less structure + data augmentation) are more or less equivalent. (Chapter 5.5.3, titled: Invariances)