I was under the impression that the W3C testing would, in fact, catch things like layout tables, etc. But maybe I'm thinking of Wave or one of the other testing tools?
Regardless, your statement that Jaws, or any other screen reader on the market, ignores stylesheets is just wrong. Using display:none and/or visibility:hidden will hide things from modern screen readers just fine. Thanks to the recent advances in things like WAI-ARIA, even dynamic content can be made to work, these days. I do almost all of my productive work in highly dynamic pages like Google Docs and Gmail, in fact. Even modern Flash players usually sort of work, at least on Windows, if you know how to move the focus to them correctly. Silverlight, on the other hand, is a nearly total failure.
Lastly, you seem, along with many other sighted designers, to make the assumption that blind people just sit down and start listening to a web page. That never happens. When a page loads, I'm generally told how many headings, links, landmarks, etc, the page has. I don't know any blind person who just starts reading a new page from the top. Generally, we just try and skip to whatever it is we're interested in. On news websites, navigate by heading (Slashdot), by landmark (Reddit), or by block of text usually get to the article fairly quickly. If I'm filling out a form, to reply to your comment, for example, I just hit e to skip from one edit field to the next. If Slashdot had a link at the top called "skip to main content", I doubt I'd even notice it. The few times I do notice those type of links, I don't use them, as the designer and I rarely agree about where the main content starts. If slashdot did have that style link, I suspect they'd put the anchor at "Reply to: Re:You Must Be New Here", even though I have 0 interest in that section of the page, as I just want to write my comment.
As a user, I feel like this kind of design is an artifact of designers who haven't memorized, and thus never use, the 20 or so hotkeys commonly used for navigating the web, never mind the hundreds of other keys needed to properly control a screen reader. And that's why I'm in favour of automated testing. All of these hotkeys depend on semantic layout, and most automated tests will at least catch missing alt-text, missing form labels, and other bad markup. They won't catch the people who use CSS styles on divs for headings, rather than the h1 tag, but at least if the alt-text is complete, it's possible to work-around that. Also, many sighted people spending an hour or so testing with a screen reader won't notice the lack of headings, or realize why that's important. In a world where the majority of pages fail even the automated tests, I'd rather push for everyone passing those, rather than taking half an hour with a screen reader they have no idea how to use, and deciding accessibility is just too hard.