It's not just the hardware constraints - it's also the training constraints, and the legibility constraints.
Training constraints: you need lots, and lots of data to build complex neural network systems. There are plenty of situations where the data just isn't available to you (whether for legal reasons, technical reasons, or just because it doesn't exist).
Legibility constraints: it is extremely hard to precisely debug and fix those systems. Let's say you build a software system to fill out tax forms - one the "traditional" way, and one that's a neural network. Now your system exhibits a bug where line 58(b) gets sometimes improperly filled out for software engineers who are married, have children, and also declared a source of overseas income. In a traditionally implemented system, you can step through the code and pinpoint why those specific conditions lead to a bug. In a neural network system, not so much.
So totally agreed with you that those are extra tools in the toolbelt - but their applicability is much, much more constrained than that of traditional code.
In short, they excel at situations where we are trying to model an extremely complex system - one that is impossible to nail down as a list of formal requirements - and where we have lots of data available. Signal processing (like self driving, OCR, etc) and human language-related problems are great examples of such problems where traditional programming approaches have failed to yield the kind of results we wanted (ie, beyond human performance) in 70+ years of research and where the modern, neural network approach finally got us the kind of results we wanted.
But if you can define the problem you're trying to solve as formal requirements, then those tools are probably ill-suited.