Can suppression of the immune system help cure chronic diseases?
No, this wouldn't happen. The logic of your proposition assumes that chronic diseases are caused by the strength of the immune response being insufficient to kill off the virus, resulting in a long-term equilibrium between the amount of virus particles produced and the amount of virus particles killed. 

In reality, this is not how chronic diseases work. Only a small amount of chronic diseases are caused by viruses. There are many which have other causes, not all of which can be defended against by the immune system - e.g. chronic depression is not curable by immune response. 

But even if we focus on chronic virus infections such as herpes simplex, they are not sustained by the mechanism you describe. They happen due to the fact that our immune system simply does not know how to get them out of their new home within our body, they are just that good at hiding during latency. So even if you could cause a strong immune response all at once, it wouldn't kill all viruses at once. 

Even if we have viruses whose serum presence depends on a constant growth vs. destruction dynamic (I don't know if these exist at all), an overly strong immune response won't be that good for the human host. Inflammation is a dangerous state, most of the unpleasant feeling you get when you are sick is not due to the virus itself but due to your own immune response, which causes the fever, muscle pains etc. associated with many acute infectious diseases. If we were to ramp this up, you'd probably get a fever high enough to cook you alive, or some other undesirable effects. 



On a side note, I also doubt the author's argument which suggests that long-ish convalescence times evolved because shorter ones would leave the system in an undesirable equilibrium. Sure, an alternative reality is thinkable where the innate and adaptive responses described by Chris occur much stronger at the beginning, leading to steeper curves and shorter healing times. But the simpler explanation for that is that 1) our immune system is not perfect, and it evolved to the highest healing speed it could achieve, and 2) an immune system with higher acceleration would have been way too costly in terms of resources available to the body. An evolutionary advantage in having a low-acceleration immune system is a strong assumption for which I see no support, especially considering that the residual immunity provided by the adaptive immune system would prevent the dynamic equilibrium suggested in your quote.