As with cryptography, it's probably dangerous to "roll your own random" if the randomness matters. The author already mentions both the point about cryptography, and the dangers of plausibility arguments about randomness: an intuitively plausible way of picking a random point from a sphere doesn't give a uniform distribution.

Playing around with distributions as the author does is surely fine if you just want to get something that "feels right", but if the applications depend on precise randomness properties, then, for example, "let's just multiply these two PDFs" is dangerous (not least because the result is almost guaranteed not to be a PDF, and may not even be normaliseable to one).

Although it surely won't be applied for random f (no pun intended), the transformation from P(f(u) ≤ x) to P(u ≤ f^(-1)(x)) relies very much on f being (strictly, in order to have an inverse) increasing—so it's even dangerous to use f(x) = x^2 if we don't know that u is non-negative-valued.