Balancing Redox Equations
1) How come I've never seen my professor's method mentioned anywhere? I've Googled, looked at scholarly papers; looked in advanced analytical textbooks, etc.


It is entirely possible that no one has published that approach! It is certainly not common in introductory chemistry textbooks.


  2) How were you taught?


I was taught (and currently teach) the "standard" algorithm you found on the Chem Wiki.


  3) Which way do you think is more advantageous?


It depends on your goal. 

If your goal is to make sure students understand all of the steps and have a good intuition for what is physically happening, then your professor's method makes a lot of sense. This is because you need to already have a clear understanding of what things like "thermodynamically favorable" and "relative acid strength" really mean in order for that approach to work.

On the other hand, if your goal is to make sure students have the functional ability to work through a particular type of problem, regardless of how well they grasp the underlying concepts, then the algorithmic method makes more sense.

And finally, if your goal is to solve a problem quickly without having to think too hard about details, the algorithmic method also makes more sense.