The problem of credit
With one neuron, training is simple. It gets something wrong. You measure how wrong. You adjust its weights a little. Done.
A network with layers is a different problem. The error shows up at the end (at the output). But the weights that caused it are buried deep inside, separated from the output by layer after layer of other neurons. When the network says "dog" and the answer was "cat," which weights were responsible? The wrong call was the result of everything that happened before it. The error is visible. Its cause is hidden.
This became known as the credit assignment problem: when the whole network fails, who gets the blame?
Researchers knew the multi-layer architecture was more powerful. They just couldn't figure out how to train it. The problem sat unsolved for two decades.