How do I get Search API to index rendered content including referenced nodes?
You should be able to traverse through the reference fields on the Search API Add Fields page. Then you want to add all the referenced text fields, so they become part of the main node. However, it is possible that Search API currently has some limitations there, it didn't show configurable fields for me when I tried something similar a few days ago. I'm sure that will eventually be fixed, though. There might be issues already.

PS: You might want to check out Paragraphs. It's a similar concept, with a much better editing experience, you can have different types of embedded content elements and edit it all in the same form. We're working on Search API integration right now and will also look into the problem mentioned above.