Personally, I see the CTO's job as being the one to advocate pristine, perfect code at the expense of business objectives. The reason is simple: if they don't, nobody else will. Of course the business side of the organization is always going to be pushing for more features at the expense of code quality. Why? Well, they're not the ones who will take the blame for code sucking.

If you give in to them too many times and end up with buggy code where trivial changes take a month, but has all the features that are needed where do you think the blame will fall? I can guarantee you it isn't going to be the business side.

The goal shouldn't be for the CTO to be the business side's technical henchman. It should be to look after the technical side of things. If a rewrite is the best decision technically (without considering the business), then that's what the CTO should advocate, but they need to be willing to work with the business side to find a solution that works for both sides.