You may have too many students if they can't space out enough:-D
Divers can use whatever they want to serve their purpose.
However, there is absolutely no reason to teach students anything other than text book. At least I am not arrogant enough to think that PADI or NAUI put in the time and research for me to doubt the teachings.
You don't exactly want to be responsible for your students getting criticized either
Well ... I question several things that I find in agency texts ... and teach many things that you won't find in them. If you go to my website and click on the Articles tab, almost every article in there either addresses something the agencies don't teach, or something they tell you to do but fail to tell you how.
That isn't to say that the agency material is wrong ... it's to say that the agencies target a specific audience, and cannot possibly cover all things in the detail that it is useful to know. That's what instructors are for ... and instructors who limit themselves to strictly to the information that's "in the book" often turn out inadequately trained divers.
With respect to the use of hands ... as with most things in scuba, my answer would be "it depends". The analogies to fish are rather missing the point. We're not fish, we're not built to propel ourselves as they do, and the legitimate question is how to move around underwater in a manner that is both comfortable and effective. In some cases, using your hands to push off is OK ... as long as you're not, in the process, damaging something. The real question you need to ask yourself is this ... are you using your hands in a manner that ingrains poor diving habits? A great deal of what you learn in scuba develops habitual behavior. Conscious decisions to scull with your hands eventually become unconscious dependencies that persist long after they're no longer useful. I'm speaking from experience here ... I was a hand-waver for years after my skills reached the point where I didn't need to be. It was an unconscious response based on years of habit ... something I wasn't even aware I was doing until one day I saw myself on video and realized I was doing it. Took me a long time and a lot of effort to stop the habit.
So why stop? Well ... because as you develop better skills, you learn to do what you were using your hands for in more efficient ways ... using less energy, and therefore less air ... which is always a desireable thing. As your skills improve, your hands become useful for other things ... like a camera, or a speargun, or a dive light ... and you won't want to be unconsciously waving them around while you're trying to use those things.
One learns best what one learns first ... that's a rule of thumb that applies to a lot of things in our life. Diving's no different. Learning good habits up-front saves a lot of effort in the long-term ... because you don't, later on, have to "unlearn" behavior as your skills improve or you move into more challenging types of diving.
Bottom line is that if you need to use your hands for something specific, go ahead and use them ... but make sure it's a conscious decision, for a specific reason, and don't get to rely on using your hands for functions that there are better ways to achieve. As with a lot of aspects of diving, there are "rules of thumb" that apply to good diving technique ... and what matters isn't so much applying them all the time so much as understanding why they exist, and when it's appropriate to use them. Try not to fall into bad habits that will limit your ability to achieve future diving goals ... or even just to help you become more comfortable with the skills you already have.
... and don't worry about what somebody else thinks ... what matters is what YOU want to do, and what skills are needed to get you there ...
... Bob (Grateful Diver)