But does that explain the transition from "innovation" to "bureaucracy"? If that claim was the case, I don't see why there couldn't be organizations that are both innovative and process-driven.

No, size explains it. You cannot be large without bureaucracy - bureaucracy means there are experts to do everything better than you can yourself. Sure small companies you can do things yourself, but you are often not the best person possible to do it - just that you are the only person.

Innovation happens at large companies as well. However not always in the same way.

I agree with the overall sentiment but perhaps disagree with the idea that bureaucracy is the result of specialization. I tend to think bureaucracy grows as a side effect of network effects. As the number of nodes in a corporate network grows, the interaction depth grows exponentially and processes help manage all those interactions. If there's a high level of specialization, but minimal interaction, there's still no need for a bureaucracy. In theory, you could have a large organization with so many silo'd departments that don't interact, there'd be no need for bureaucracy.