This method only works because the boxes have the same number of balls. To calculate probabilities by counting outcomes you have to start with outcomes that all have equal probabilities.
The more "intuitive" way of generalizing it (integer counting) does not work in general. I think the example is likely to be misleading to anyone who doesn't already understand this stuff.
As suggested by the footnote, the reason the given example can be solved elegantly is because of the symmetry.