Okay, I created a parent / child relationship between the spine joints and the leg and arm joints (as is usually done) and all seems well. For some reason, I remember the instructor mentioning that one should never interrupt the flow of joints with other joint chains, but going over the videos again, I can’t find the comment, so maybe I just imagined it.
Also, I thought using a spline IK chain might require a special constraint to connect the legs and arms to the main spine, but it appears to work just fine without them, so I assume all is well…