I'd just like to note that instead of creating additional animosity between SVMs and deep nets, you could use both together. SVMs with hinge loss can be Yet-another-layer (tm) in your deep net, to be used when it provides better performance.
One nitpick though, ConvNets can absolutely be used to do "thinking" and more than just feature extraction. For example, fully convolutional networks can be extremely competitive with FC-layer based nets.
https://arxiv.org/abs/1605.06265
http://papers.nips.cc/paper/5348-convolutional-kernel-networ...
You also want to be able to train the DNN on your unlabelled data and the SVM on your much smaller labelled set.