This may not be obvious, since those startups are typically "agile" in the buzzword sense.
The cost comes when the universe of possible change is constrained to the point where the right questions aren't being asked because the answer would be "no we can't do that". So the startup enters a dark age in which scientific thought and rational problem solving isn't really relevant. Often at this point the team starts using Scrum to force the business to plan better, when the core problem is that the existing infrastructure is not able to support the needed experimentation fast enough, making it useless.
This can happen at various stages of funding, since sometimes a series A or B can occur without solid product market fit, or the company may be really good at growth in spite of having a promising but mediocre product.
So I view the CTO's job as preserving the ability of the business to behave scientifically and boldly until product market fit is found.