One viewpoint is that the basic barbell lifts train the "core" sufficiently. And indeed it seems reasonable that anyone who can squat and deadlift large weights have very strong core muscles.
Dr. Stuart M. McGill was a professor at University of Waterloo for 30 years where he lead a research clinic that investigated back pain. He has published 240 peer-reviewed scientific journal papers. Brian Carroll is one of the worlds best powerlifters. In 2013 he was told by doctors that he could never lift again due to serious back injuries. After training with McGill for 10 months he completely recovered. He is now working with McGill.
In this video they consult Layne Norton who is suffering from backpain. Norton is himself an extremely accomplished powerlifter and bodybuilder and is also very smart and knowledgable about strength training. In the video (at 7 minutes) you can see how Norton struggles with the simple bird dog bodyweight exercise. This is very odd and seems to contradict the view that anyone who can squat and deadlift large weights have very strong core muscles. Norton can deadlift 322 kg and squat 303 kg at 93 kg. He should have plenty strong core muscles.
The same thing can be seen in this video (starts at 7 minutes). Here Dave Tate, PR: 335 kg deadlift and 424 kg squat struggles with the same bird dog exercise.
What is going on here and what implications should this have for ordinary people (who do strength training)?
One thing I notice is that in basic barbell training the cross pattern (left leg to right arm and vice versa) is never trained. In athletics and everyday life this pattern seems very important. It us used when running, shotputting, boxing etc. I wonder if what we see is more a problem of coordination than lack of strength. Perhaps Norton and Tate is so used to excerting force in both legs simultaneously that they struggle to "switch off" one leg and still excert force trough the other.
The functional movement screen (FMS) includes the bird dog and the same exercise but with arm on same side as leg (ipsilateral). According to this article 75 % out of all pro athletes struggle with these excercises. Knowing that the purpose of the FMS is to identify athletes that are in risk of injury this does not sound good. Granted the people administering the test have a vested interest in finding lots of shortcomings, but still.