I don't understand why they would want people to exercise to anaerobic threshold.
Because it is the most interesting phenomena if we wish to understand the altered effort perception in CFS patients. I agree that this might not be so useful in terms of management...
Amongst healthy people and patients with various conditions (other than breathing restricted conditions such as COPD), the ventilatory threshold/anaerobic threshold directly matches the increase in effort perception on the Borg scale during a CPET. In CFS patients who have a reduction in their anaerobic threshold due to PEM, guess what, a change in effort perception directly mirrors this reduction.
There is a however myth that it is pushing past the anaerobic threshold that causes PEM and that keeping your heart rate down below this level will somehow prevent PEM. I have had severe PEM after doing prolonged low intensity activity (fixing my electric bike), but avoided severe PEM after exercising briefly past my anaerobic threshold while pedaling on said bike (heart rate 160+) and then resting. Occasional brief spurts above the anaerobic threshold is how you can maintain or even increase VO2max without doing much exercise overall - you only need to do less than 10 minutes of exercise one day a week to maintain an above average VO2max for one's age and bodyweight.
I may be wrong but my understanding of an anaerobic threshold is that it is the difference between running a 100 metre sprint and running a 1500 metre distance race. Running a 1500 metre race competitively, which most of us get nowhere near, is still under the anaerobic threshold, because you can only continue for five minutes if you are using aerobic metabolism. If you exceed your anaerobic threshold you get badly out of breath and there comes a point pretty quickly where you simply cannot continue. If I am ill I find it hard to do normal activities like going up one flight of stairs but I don't think that has anything to do with anaerobic threshold. I am not out of breath, I just cannot get my muscles to do what I want. I think that has more to do with cytokine signals.
If you want to be fast in the 1500m, you will be regularly exceeding the anaerobic threshold during training ("speed training"), so that you can improve your performance at the threshold...
I don't believe there is any serious evidence that the effort perception you are speaking of is due to cytokine signals. However there have been a few interesting studies which have tried to investigate what could influence such signals, eg does infusion of adrenaline alter VO2Max or effort perception? (answer: no)
https://www.ncbi.nlm.nih.gov/pubmed/16215769
Or am I mistaken in thinking hitting the wall is the same as hitting one's anaerobic threshold?
Yes, it is quite different. Hitting the wall is due to glycogen depletion, not merely exceeding the anaerobic threshold.
This does beg an interesting question though...
"The effect of exercise induced glycogen depletion on the lactate, ventilatory and electromyographic thresholds."
https://www.ncbi.nlm.nih.gov/pubmed/9190123
"Effects of glycogen depletion and work load on postexercise O2 consumption and blood lactate."
https://www.ncbi.nlm.nih.gov/pubmed/533743
"
Effects of glycogen depletion and pedaling speed on "anaerobic threshold"."
https://www.ncbi.nlm.nih.gov/pubmed/6809718