Simple answer: No one set out to solve big problems (computers) like that. If you went back in time to make a transistor, you'd be solving the issue that was the problem back then. Generally it was how to recreate and transmit radio signals for further distances. Ironic, considering that was an analog endeavor.

Computers emerged out of the discovery of correlations with logic. Many, many incremental steps to make things better, faster, and more reliable slowly developed into what we see now. Transistors the size of your hand had fundamentally different purposes back then than the millions found in the size of your typical processor, even if they do relatively operate the same.