The hip flexors do not cause initial hip flexion.

The hip flexors are not responsible for pulling/flexing the swing leg forward in gait or running. The psoas is a mere swing phase perpetuator, not an initiator.
For about 2 decades we have been saying in our lectures, posts and podcasts that it is the reduction of the obliquity of the pelvis during gait from various other tissues and biomechanical events that causes leg swing, meaning the trail leg is brought forward in swing largely by the abdominal muscle linkage to the pelvis (and other loaded tissues) that is responsible for forward swing of the leg. It is not the hip flexor group that does this hip flexion action. Thus it could be considered foolish to train the hip flexors to be the primary swing drivers. Here is another supporting piece of research.

"These experiments also showed that the trailing leg is brought forward during the swing phase without activity in the flexor muscles about the hip joint. This was verified by the absence of EMG activity in the iliacus muscle measured by intramuscular wire electrodes. Instead the strong ligaments restricting hip joint extension are stretched during the first half of the swing phase thereby storing elastic energy, which is released during the last half of the stance phase and accelerating the leg into the swing phase. This is considered an important energy conserving feature of human walking. "

Dan Med J. 2014 Apr;61(4):B4823.
Contributions to the understanding of gait control.
Simonsen EB1.