Browse the Site Settings of a site and the Site Templates link is missing from the page. The image on the left shows the Galleries section with the missing Site Templates link. The image on the right shows what it should look like.
My first thought is a feature has a HideCustomAction that is hiding the link from the page, but comparing a working and non-working sites confirm that the same features are active which makes this unlikely. We then browse the URL to the Site Templates gallery: http://site/_catalogs/wt/forms/common.aspx. The page renders, but appears to be missing the List View web part that displays the items. You can tell as the New, Actions, and Settings items are missing [See image below]. Even if you had read only access, you should get the Actions menu. When this happens, the page is no longer considered a view for a list.
If you open the site in SharePoint Designer and look at the properties of the Web Template library, you see that the Default View Page dropdown is empty. This dropdown is set to Common.aspx on a working library.
You get into this situation by browsing the Site Template Gallery, then going to Site Actions, select Edit Page. From here, you can delete the List View web part from the page. You could also open the site in SharePoint Designer, open the Common.aspx page and remove the web part that way as well.
To fix the problem and restore the link, you can browse directly to the AllItems.aspx page. Unless the user removed the web part from this page, you should be able to get to the Library Properties from this view. From here, you can create a new view and make it the default. Once you set a new default view, the link will return to the Site Settings page.