For building custom connectors to sources like RESTful APIs, web data connectors are the way to go.
The new Extensions API is currently targeting custom extensions inside a dashboard. These extensions can interact with other zones in the dashboard and, from within the extension's web application code, they can interact with any other system they can get to, such as writing back information to an inventory management system, sending a request to a Natural Language Generation service, or simply loading up D3 to draw a network diagram. There are tons of possibilities. I recommend going to tableau.com/developer to learn more about them if you are interested.
For custom connectivity, though, web data connector is the path for you. If that service provided an ODATA endpoint, that would also be an alternative (no code) or if they happen to have an ODBC driver (unlikely but possible) also a no-code option but if not, WDC. Resources for WDC are also available at tableau.com/developer.
Many thanks Lee, that was the conclusion I had come too so very much appreciate the confirmation.
All the best, Ben.