The observation that an average sounds flat is a good one. The relative phase of sounds in each channel could do weird things and cancel each other out when averaged. However, most modern music is mastered in such a way that the stereo channels are highly correlated, so averaging is usually the best way to get a mono version.

That said, the solution in the article isn't really doing much...

given two tracks: L and R

  side channel = (L - R)/4dB
  mid channel = (L + R)/4dB
  final = (L - R)/4dB + (L + R)/4dB = 2L/4dB ≈ 1.26L

Since dB is a logarithmic scale, 4dB corresponds to around 1.58 times power. 2dB corresponds to around 1.26 times power. Audacity may be attenuating it further in each of the "Mix and Render" steps depending on how it's summing is implemented.

In the end, it looks like you'll up with just the left channel. And as a practical solution to the original problem, this isn't that bad of a result!

Edit: formatting

Isn't this exactly the same as what you were doing before, averaging the channels and then amplifying?