Neural networks absolutely are designed. Network architecture is one of the primary innovations in NNs over the years, and transformer architectures are the biggest development that enabled modern LLMs.
In addition, behavior is designed indirectly through human reinforcement.
The individual weights aren’t designed themselves, but there is a ton of design that goes into neural networks.
I'm admittedly out of my depth, but think the point is the current state of architecture is one of rapid and volatile iteration. There isn't really a comprehensive "design" because each generation is building (and in some cases rebuilding) upon the previous generation.