I'm not any kind of expert on POSE (beyond website research), but I did well in first year physics. Here's my 2 cents.
I think this idea of falling forward only makes sense figuratively. It's a way of getting you to think differently about technique, that enables you to train certain "new" techniques in phases of your running gait (i.e. pulling your leg up, rather than pushing off), and de-emphasize others, to bring about the desired changes in running form. So thinking about pulling your leg up, may mean you still push off, but just push off less.
The physics have been analyzed to death before, in lots of other threads, but basically, you cannot escape the simple fact that what goes down, must come up. If you fall down at all, due to gravity, something has to bring you back up, in order to fall down again.
But POSE runners talk about falling forward, not down. When you think about your running, at some point your foot is stuck to the ground, held in place by static friction, and your body has a forward momentum, and this generates a torque (or should I say angular momentum?) around the planted foot. Once the center of mass is in front of the foot, gravity can contribute to accelerating the rotation, only if the center of mass falls (i.e. potential energy converts to kinetic energy). But if you want to repeat the process at the next step, one of your feet has to bring the center of mass back to the same initial height. I guess this typically happens when you push off, as you become slightly airborne with both feet off the ground. If it doesn't, then the landing foot will have the duty of restoring the center of mass to the right height. You only have two feet, and they are the only thing in contact with the ground as you run (unless you really fall).
I think POSE can help you, only if there is something wrong with your current form, and learning POSE gets you to correct it. If you are running like a pogo-stick, POSE may get you to be less bouncy, and save some of that energy for going forward.
POSE can help correct certain kinds of injuries, as it moves the stress of running from your hips and knees, to your ankles and calves. When you learn POSE, you usually need to do drills to strengthen your ankles and calves.
For me, the biggest problem with POSE, it's that it's only available to you through an info-mercial type weekend seminar.
Keep in mind that Haile G and Usain Bolt likely learned proper form without Dr. Romanov.