Your history is a little mixed up. LeNet-5 had 7 layers, for example (late 1990s). Regardless, neural nets fell out of favor because they were seen as overcomp

Your history is a little mixed up. LeNet-5 had 7 layers, for example (late 1990s). Regardless, neural nets fell out of favor because they were seen as overcomplicated and even though they achieved competitive accuracy (yes, even when they were out of favor they were still competitive in performance to other methods), you could get similar performance with simpler methods like SVMs.