So, I did a lot of reading on the topic. The bottom line seems to be that, while TrA training seems to be effective for some outcomes, it is not more so than other treatments. There are some interesting studies that show that these exercises can "normalize" the activation pattern for the TrA, but there's a severe lack of studies comparing the TrA approach to other forms of PT (and where they exist, the differences are mostly not significant!).
Please note that we're talking about chronic LBP, the systematic reviews do NOT recommend to use the stabilisation excercises for acute LBP.

concerning the article in the initial post:
There is some oversimplification (isolated TrA training is not advocated by the "core stability" proponents) and some error (it is not about strengthening, but about recruiting the core). but i like the critical appraisal of what seems to have become a dogma.

In the end, you can say that CS is one valid approach to the topic, but in no way a panacea.