CRO or Conversion rate optimisation is where SEO was a 5/6 years ago. Every body want its but not everyone really understands what it is. Usually it is explained to C-levels as way to convert more users by running experiments (a lot of people say tests but I prefer this term to avoid confusion).
This is rather simplistic because if I see that a page is broken on a particular type of browser or device I will get it fixed with out experimentation. The same with back end systems if there is a issue that is causing problems such as a poor returns process then it should be fixed.
Also another big issue I see in CRO which is a legacy issue from the early days of SEO and PPC is every one is chasing after the easy wins or the low hanging fruit if your prefer. Yes they require little effort and yield results but only 10-15% of the fruit is easily reached. The vast majority will take effort but will yield vastly greater returns.
So how do you get buy in for CRO above and beyond basic button changes and headline tweaks? You have to let the wookie win at first and prove the benefits of CRO through small test such as headline and CTA changes but then you should show that if the headline, copy and CTA are all aligned to the same goal then the returns are greatest. Then you show if the design, copy and ads are all aligned then the returns are greatest.
Once you have some good lifts in conversions then you can use the learnings from your tests, qualitative and quantitative data and try a fully redesigned page against the existing one.
It takes time and research before you can safely and confidently launch a brand new page design on a site. Yes you can do it without incremental tests if you have a solid analysis and research to back up your findings.
Either way you need to show your client or boss that the low hanging fruit is a useful acquisition but it will not transform your business and that is ultimately the goal of every CRO professional.