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.

If you're willing to watch some YouTube videos, ben eater has a playlist on exactly this. From how semiconductors work on up.

A transistor is just a switch that can be on or off. The original computers used giant valves that could be opened or closed.

They are an electronic component in their own right and not specific to computers.

Combine them with resistors and you can make logic gates (and, or, not, etc) combine these and you can make even more complex things.

So a transistor on its own can either hold a charge or not. It doesn't do anything useful until you combine them with other things.

I would look up videos about how microcontrollers work as they are way simpler than CPUs but the same principal.