I’ll be honest, I haven’t had any experience with the WP Touch plugin so I can’t speak to how it’s actually created or implemented.
The Staff section is built on a custom post type. When these post types were first introduced, they did not have a way built to display an archive of them. So my first method to display them was via a page that used a custom query to display the results.
I had several requests for the ability to use multiple listings, so I added support for a custom taxonomy (staff group) so you could create multiple lists that were separate. This also worked well because even after WordPress added support for custom post type archives, they did not include them in the navigation builder (you have to add a custom link to them). Custom taxonomies are included by default.
My guess is that the theme that the plugin uses doesn’t have the support for the custom post type/custom taxonomy in it directly. I didn’t see mention of it on their website. Since you can create a custom theme to use for the mobile site, you could probably add the support there. That’s where I’d start at least.
Let me know if you have trouble and enjoy your Thanksgiving.