The horizontal component of the reaction force responsible for forward movement of person while walking is regarded as frictional force. But when a person pushes the ground backward, the supposed relative motion is person moving forward. Then the frictional force should act backward opposing that relative motion. But then why is that horizontal component of reaction force that is acting forward regarded as friction?
The frictional force is static. Your foot on the ground has no relative movement (unless you slip) with respect to the ground. Thus the frictional force does not oppose movement, but the action force by which you push on the ground. Your feet push backward on the ground, making the reaction forward.
But this frictional force is perfectly balanced otherwise it would be your feet that accelerate and not your body. The frictional force is necessary to make your feet static, so that when the rest of your body pushes on its feet, they will not move backward, as they do for example on ice.
When runners accelerate after the start shot, their foot press a fixed device backwards. In this case, most of the reaction is normal force.
What is important, in this case and in walking, is that the body is momentarily accelerated by the reaction force, no matter if it is fully static friction, fully normal force or a mix of both.
What opposes movement is kinetic, not static friction.