Instead of just using good markup in the first place?

Good markup can't handle it all, text-only mode allows for stuff like having a transcript instead of a video with a link to a transcript. Text-only mode is even available on the new site:

http://www.whitehouse.gov/accessibility/

"In addition to the features that make the graphics version of the site more accessible, a text-only version provides added convenience for users."

I'm not sure how to access it though, maybe that feature hasn't launched yet.

I'm going to disagree. Let me share some background, I spent several years helping to write the Web Content Accessibility Guidelines 2.0 spec. The original WCAG 1.0 as well as Section 508 specify an alternate text only site as a viable method.

Real world experience shows us those sites quickly fall behind. Text-alternatives are necessary for multimedia content, text-only sites as an alternative to good markup aren't.

Fall behind in what sense? Obviously, if the multimedia content isn't transcribed quickly enough, then that makes sense, but I would also guess less consequential than if a given page weren't transformed from 'marked up' to 'text', but in the latter case, if the backend is dynamic, and just changing the output format, why would it fall behind?

People just don't want to maintain 2 sites. Extra features and sections often don't get updated to the text only rendering.

I guess my point is that screen readers like (even prefer) well written markup. A text-only site is a kickback from when they couldn't and the accessibility guidelines were written to deal with that.