There may have been a mix-up in discussing the Server API at rstudio::conf, or perhaps I am missing something. That is definitely our vision for where the Server API will someday be, but the Server API made its entrance in Connect 1.6.0 (released in April 2018) and is currently in its early days. The Server API endpoints are backwards compatible, public, and documented.
As you have suggested, there is an internal API that is undocumented, but we do not recommend or support trying to use it, make no guarantees about its behavior or backwards compatibility, and ultimately would recommend waiting until we add content listing (the feature that you are requesting) to the Connect Server API.
In the meantime, we would love feedback on the Server API features that we have exposed, if you have any need to utilize them! Stay tuned to the RStudio Blog (blog.rstudio.com) or subscribe to stay up to date on latest releases and the features we continue adding to the Server API.
At the time of rstudio::conf, the Server API did not exist yet, so I expect that discussion was occurring around future vision, may have been discussing the internal API, or there was a mix-up discussing API assets like Plumber REST APIs or Tensorflow models. I for one am super excited about the Server API's future, and am encouraged to hear others excited about additional features.