Engineers are the only ones who can advocate for pristine code without tradeoffs, and I would argue that great engineers care enough about the business to not do that. Really great engineers just build the pristine code in the same amount of time regular engineers would have taken to get a simple version almost-working.
I actually like the idea of "technical henchman", though!
This has been key in my experience. I'll build a feature that is flexible, clean, well factored, and completely documented in the same time it requires someone else to make a half working mess.
CTO's as I'd like to understand them bring the bar as high as they can within business constraints. Doing any less is accepting mediocrity. Is that something you'd want in an executive leadership role?
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.