Stevie gave me the heads up, so I decided to pay a visit.
For Stevie... This is an oft-discussed subject in these Forums. Given my academic training (while practicing martial arts) in systems physiology with a specialty in cardiopulmonary systems, I've had time to ponder it and even indirectly relate it to my dissertation research. I've also had the good fortune of walking into martial arts as an athlete (track and baseball) and studying other styles (Nippon Shorin Ken, Goju Ryu, etc.). So yes... I have some opinions.
I'm at peace with my own approach to breathing. I'm not a student of any single way, but rather use whatever breathing seems to match what I'm doing physically.
The first and most important thing to remember is to breathe. That may sound stupid, but it isn't. Under stress where many get deer-in-the-headlamps syndrome, some will hold their breath. Most breathing methods are good merely in the fact that they get the practitioner to breathe when doing martial movement. Everything from that point forward is icing on the martial cake.
If one lives by the principles of Uechi, then one considers that over-extension is rarely done. Much of Uechi Ryu is exactly like the same biomechanical principles we see in any other sport. The biggest difference is we do everything smaller because over-extension can create vulnerability and speed (glare in the eyes with fast hands) is a prime directive. So yes... all that you learned with Peter Kellog is true. And the principle of power breathing Van talks about is a good one. Marry the two together, and you get something unique to the style.
Or not... Nobody says a Uechika needs to be bound by the style. The principles are there to use, or not. There is no one "right" way.
Just as one can get all obsessed with Power Breathing, so too can one go off the deep end with Kanei's more passive approach to breathing. I think the correct way to think of it is the way another of Van's friends likes to think of it. Instead of thinking of breathing driving the power, we can think of the movement itself breathing us. We can "be breathed" by our whole-body mechanics. This fits with the description of Uechi Ryu as "hard on the outside, and soft on the inside." If we don't hold that breath when we move, breathing will naturally happen. Then you can go with it and dial it up as it seems and feels appropriate.
You want some science...
Well the first part is the stupid-simple part I mentioned. Don't forget to breathe. Power breathing gets you to breathe with movement. The between-breathing gets you breathing in-between movement. Both are important. Both should be practiced so you'll do it under pants-pizzing stress. Everyone is right here, and everyone wins.
The science of power breathing in martial arts isn't about aerobics. Cross country running is about aerobics. Basketball is about aerobics. Rowing is about aerobics. Martial arts - particularly the striking arts - is primarily an anaerobic activity. So obsession with optimal O2 transport to tissue is a waste of time. Whatever energy you need to beat the beast comes with the phosphocreatine and glycolytic energy pathways. That energy is in the bank at rest, ready to be used. Anything that comes from aerobics is long-lasting, but a fraction of what anaerobic metabolism can generate.
The best way to think about it is to think about a properly-inflated basketball. Too little air and you don't get an elastic collision when you bounce the ball. Too much and you do to the basketball what my son did to my riding mower tire a few weeks ago when he overinflated it. There is a right pressure for that basketball. There's a right pressure for the tire based on the riding you plan to do.
Whether you realize it or not, power breathing involves restricting the exhale a bit. Restrict it maximally (while still breathing) and you get a kiai. Restrict it very little and you get smooth movement. Do something in-between and it's optimal for some particular kind of technique.
Whole-body movement and whip-like power (sequential summation of motion) involves generating most of the force with the big muscles (legs and hips), passing that power through the torso, and having it radiate out to the periphery. How that energy passes through the trunk depends a lot on the viscoelastic properties of that trunk. And that depends on the intraabdomenal and intrathoracic pressure, which is affected by power breathing or something else. It's the tire pressure thing. You want lower tire pressure to grip the road, but higher tire pressure to carry a heavy load or go high speed. It all depends.
The best athletes don't over-think this. My recommendation is to carry this into the weight room, and put it to practice. What I like doing however is getting beyond the stupid-simple stuff of bench and squat. Try multi-part exercises like the Olympic clean-and-jerk, where one needs to use more than one breath. Do ten of them. Another of my favorites is the Turkish get up. Again... do ten of them. Find out how to weave the inhale (very neglected part) into the exhale. I can't tell you how many times I've had to shake my head in discussions with the exhale-obsessed because they fail to acknowledge that yang needs yin. Restricted inhaling (sniffing) works like restricted exhaling (hissing), only to load before unloading. With multiple movements, you need to get it all to balance out, unless you're fond of practicing red-faced. And that by the way isn't healthy. What you'll find is that you exhale, you inhale, you pant in places, and you even briefly hold your breath in places. See what works and then think about why. You start with these good ideas and then put them to use with your car in gear. With the feedback of real weight (and various levels of weight from light to heavy), you'll find the right paths. When we're swinging at air we can lie to ourselves. The weight doesn't lie.
Listen to what folks are saying here. Everyone's got it right. Everyone's got a piece of the picture. All we have to do is put it together in a way that makes sense in our practice.
My opinions, of course, worth what you paid for them.