I am specifically complaining about WordPress. Anyone could make a server side dynamic application that applies proper caching headers to work with a CDN. Anyone could, but WordPress specifically did not. It's an uncacheable mess by default and caching plugins just barely make it useable.
Using a static site generator is basically just a way of ensuring that the pages are properly cached by the brute fact that they've been dumped out on disk. It's not strictly necessary for a well designed system, but it raises the floor because even in the worst case, the pages are static files.
For many public facing sites, dynamic applications aren't strictly necessary. If you're just hosting a PDF of an Excel spreadsheet of permitted job categories, you don't need a dynamic application. Again, a well designed app would already be hosting this through S3, but you can't trust things to be well designed when made by a contractor with no technical oversight.