I haven't read the book, but from your summary it sounds like the argument is that for illnesses that we don't yet understand well enough to cure or even have a potential cure, we should use a different word. That sounds pedantic.

There's many non-mental diseases that we don't yet understand fully enough to know how to cure completely. There are many diseases where there are varied causes of the disease, where we are still learning these causes, and still learning what could be a cure. Should all these not be called an illness too?

The disease like this that I'm most familiar with is Cystic Fibrosis. 35 years ago, doctors had no idea what caused it. Now we know it is genetic. However, the disease is caused by many different possible combinations of gene mutations and we're still discovering more that can cause it. There's currently over a thousand gene mutations that cause Cystic Fibrosis. While a lot of progress in slowing disease progression has been made, there is no cure. We're not even certain about what would be needed for a full cure as science is still learning about the full extent of the disease. Cystic Fibrosis does not meet your criteria for an illness either. Should we not call it as such? I think that would be absurd. It is clearly an illness.

Just because we don't fully understand a disease yet doesn't mean it can't be called an illness. If you want to come up with a new word that can clarify the fact that we don't fully understand it yet, I guess that's fine, but I don't like confusing people by trying to deny the common understanding of a word.