By the way, hotdoc works on any platform where Python runs. In particular, once you've got Python running, pip install hotdoc installs it for you even if you don't have packages for your distribution.
As for API documentation vs Library documentation it just seems like HotDoc is for Library documentation. (Even though it's the tastiest API documentation system) From their examples they are just showing Library documentation sites?
Docsgen is inspired by API docs like DO and PayPal where you have text on left and example request and response on the right. Example: https://developers.digitalocean.com/documentation/v2/
I have trouble finding examples of API documentation sites that uses HotDoc, feel free to link some and i will check them.
As for API, I now see what you mean, you use the term API is a different meaning than I do (remote API vs local API), hence the misunderstaning. No, I don't think hotdoc supports side-by-side documentation and examples at the moment. I think it can be extended to do so, though.