yes and no, i agree that core is important, what ever people are actually defining as the core, but not as some seem to suggest more important than any other muscle grouping, its more or less useless unless you have equally developed shoulder gurdle/ arms and legs/ glutes etal. AND the body is so interconected, that an over developed muscle grouping is every bit as bad as an under developed one.
For instance a lot of lower back problems have little to do with a weak lower back, but come from under developed glutes and hamstrings,.
strengthening you lower back further will make the problem worse not better. That why i favour whole body movements, that naturally develop the whole muscle chain in the proportions that it requires. Rather than just trying to isolate specific core muscles, that then cause you imbalance problem elsewhere.
at the end of the day, these are by and large postural muscles
, and their first and prime responsibility to to give you correct posture. The,abs should keep your guts in and you back should hold you upright, if they are not doing that, they are failing. And you develop that by walking around upright with you guts held in, you don't need fancy core exercises
And they need to be in balance with each other to do that, as well as in ballance with every other group to give you both posture and movement against resistance