When the new CEO comes in and wants things done immediately in the big company way, it's going to feel like the new guy is saying he was doing everything wrong. Further, the actions he was taking will be perceived by others in the company and by new management as his identity rather than as a rational response to the circumstances of the early company.
A smart and observant person in such a role might come around over time naturally. He or she would notice that what worked early on isn't working as well any longer and would adapt. That may even be better for the company than going overnight from "small company mode" to "big company mode".
Or the person may not come around. Either way, it's likely change will not be perceived as fast enough. Difficult problem for all parties.