First of all, just because you don't end up flat on your face after every step doesn't mean you're walking without falling. The falling motion, when it comes to gait, is actually quite small, not the head over heels face plant you're thinking of. The problem comes from the way we move forward. If you're lifting your leg out in front of you to walk like this:
Either you fall, or you bend your other leg to get your front foot on the ground, but then you'd look like you were supposed to be in a Monty Python sketch. This is a really high impact way of walking on lots of different body parts. First of all, your center of mass is constantly thrown over the front of your body. This overloads your knee and the front of your foot, since they weren't designed to carry the weight of your body from the knees up.
The second problem is that this motion is causing you to pivot on your foot. Instead of your body just gliding forward, it is pivoting from a single point (your foot on the ground).That means your big fat head has to travel a further distance than any other part of your body! Not only does it have to travel further, it also has to do it in the same amount of time...which means it has to go FASTER. That's right. Your fat stupid head has to travel further and faster through space than the rest of your body.
Something has to eventually stop you from pivoting, though, or you'd end up face planting.
Luckily for you, your other leg is bent at the hip and knee to catch you every. single. time. Ouch. So, the leg breaks your fall, but something still has to stop the acceleration of your fast, monumental head. Otherwise:
So you're thinking "Thanks, GAIT keeper" (haha, get it?) "how the eff am I supposed to locomote then?" Well, simply put, you're supposed to put your weight on one straight, vertical leg and push yourself forward. Then you land on your other leg, which stays vertical beneath you, and that becomes your straight, vertical weight bearing leg which you then use to push yourself forward again. This way, your weight is always stacked where it should be, and your whole body moves through space at the same speed!
A final note about walking: you cannot walk on a treadmill. Walking requires that you push your weight forward off a fixed ground. On a treadmill, the ground is already moving, so you have no other choice than to....*drum roll please* lift your leg out in front of you and fall down onto it. And that, my friends, is not walking.