In the model of 'eating their own dog food', government agencies should not just offer the public APIs of all their front-facing content and data, but consume those APIs for their own products. This is not only more efficient but will reinforce the use of APIs as well as promote a better feedback loop whereby agencies will find and correct errors and improve the format of their public offerings.
The FCC has opensourced a Drupal module that turns all of the web content on its website into a publically available API. More so though, it built the redesigned FCC.gov by consuming those same web services. This model makes for a cheaper and more efficient development process and lays the groundwork for future returns on investment. Encouraging this practice further in government will not only result in more information being offered to the public in productive and desirable ways but will also encourage the elimination of duplicative efforts and promote greater data quality efforts. It also promotes the adoption of open standards and best practices across agencies.