Am I wrong if I think that Web Designer/XaraXtreme is a tool that can make a good personal website, but not good for doing a website to a client?
That's not really the distinction. Xtreme is good at fixed, graphics-based brochure-like sites, and for rapid prototyping. It's poor at dynamic, content-based, interactive sites; that's not really what it's made for.

Whether the site is personal or for someone else is not much of an issue.