Do you as a runner, or a helper of runners, ever experience low back tightness on a run, particularly a long run ? These muscles can fatigue just like any other muscle; we just generally do not think of them in this manner. Be sure to endurance/fatigue test your runner's lumbar erector muscles. Sure, we need a strong durable core, but in an endeavor like running, where there are constant acceleration/deceleration as well as impact loading forces over long distances, we need these muscles to be just as durable as any other.
And, do not forget that we have to control the constant anterior and posterior pelvis oscillations in the sagittal plane (APT and PPT) that correspond to the acceleration and deceleration phases of gait.
"Previous EMG studies of running have shown the lumbar extensor muscles to be active at foot contact and during early stance (Thorstensson, Carlson, Zomlefer, & Nilsson, 1982). This early activation of the back extensors will act to limit forward flexion of the trunk as energy is absorbed in the lower limbs and the CoM decelerates. During this deceleration phase, the thorax moves into forward flexion and there is a small corresponding posterior tilt of the pelvis as gluteus maximus acts to extend the hip."
S.J. Preece et al. / Human Movement Science 45 (2016) 110–118