By not designed I'm talking about the synaptic weights

Still designed by humans. The loss function, backpropagation and all other mechanisms didn't just appear magically in the neural network. Someone decided which loss function to use, which architecture or which optimization techniques. Only because it takes a big GPU a lot of number crunching to assign those weights, it doesn't mean it's biological.

In the same way, a weather forecast model using a lot of complicated differential equations is not biological. A finite element model analyzing some complicated electromagnetic field, or the aerodynamics of a car is not biological. Just because someone around 70-75 years ago called them 'perceptrons' or 'neurons' instead of thingamajigs does not make them biology.