So apparently I can't select both the RoG and Shipwrecked DLC when I start a new game.

I remember seeing in multiple places that when they changed the idea for shipwrecked from a standalone game to a DLC, the idea was to let it act as additional gamespace. I can't find anything that says that feature idea was dropped/removed, but it doesn't look like it's in game since i have to pick which expansion to use.

Does anyone know whether that feature was dropped, or if this is a bug, or what's up?

Is there a way to start a regular game and progress to the other two during the course of it?

No, there isn't. Because Reign of Giants is an integration of content in the regular game... it just wouldn't work. Shipwrecked is essentially a different game, and you can only move to and from it as mentioned above.
edit: aww look at that, I'm slowly getting my own stalker gathering just like you :)