I believe the same thing you do; that proper semantic markup improves the UX of the website.
Nevertheless, to directly answer your question: I do believe you are misunderstanding (or probably more properly miscommunicating) the scope defined between you and the client.
So if I might take the liberty of doing my best to answer the question I believe you’re actually asking ("how do I communicate the UX benefits of semantic markup?"):
Since markup seems to many customers as "code" (a necessary evil, not the thing they're paying for), you need to focus on the way those semantic markup principles and accessibility improvements can affect the user's actual experience. You might, for example, like to get a copy of JAWS (on Windows) or set up VoiceOver (on Mac) and demonstrate how applying a strong page hierarchy using headings allows for much simpler page navigation for the blind, and how
alt tags make content images meaningful for them.
When I'm communicating the benefits of microformats to customers, I like to use an extension to demonstrate how search engines, future browsers and other content consumers can benefit from proper markup of things like hCards and hCalendar.
You can also demonstrate the document outline using extensions to show how the user's actual interaction with the page can be improved using proper outlining semantics.
The last suggestion I have is related to link types such as
link rel="next": there's a Firefox extension called Link Widgets that provides in-chrome navigation of pages in a site. You can also mention with these that browsers can improve the UX through prefetching when you include these links.