No, sickness behavior does not cause the symptoms. Symptoms are caused by biological processes . Not by behavior.
I think you may be misunderstanding what is meant by sickness behavior. The theory of sickness behavior posits that an infectious illness will trigger off certain behaviors, and what we perceive as symptoms (eg fatigue) are actually behavioral responses that, like automatic reflexes, are activated by the infection, and that those behaviors are there for specific purposes.
As mentioned, I question this theory, and wonder whether fatigue may just appear incidentally rather than purposefully; but that's the theory as it stands.
Much of human and animal behavior is automatically kicked off by various factors, even if we like to think that we are completely in control. For example, if you see a member of the opposite sex that you find attractive, that stimulus will tend to trigger various behavioral responses, both in humans and other animals. Millions of years of evolution created those responses.
I am interested in the idea of sickness behaviour being purposely designed by evolution. It sounds rather teleological to me.
If you believe that the forces evolution created all forms of life on Earth, which I am sure all of us here do, then by extension, it follows that evolution must have created all the intrinsic behavioral responses (those behaviors which are hardwired into the brain).
The behavior we learn from our environment is different though: if you have learnt to behave in a certain way from your cultural and upbringing experiences, then that's not intrinsic or hardwired, so would not be created by evolution.
But I don't think we need to get hung up on psychology here, as ultimately it is not an issue of psychology.
The basic question I think is whether the symptoms / behaviors / actions or whatever you want to call them that occur during an infectious illness like the flu, and the biological mechanisms that drive them, are the same biological mechanisms that drive some of the symptoms of ME/CFS. In other words, is ME/CFS just like a flu that never goes away?