It is a sort of automatic reflex when encountering stuff like $x^x$ in such contexts like here in mathematics to take logarithms. Working with logarithms is usually much simpler and easier to understand when having variables in both the exponent and the base, both for technical reasons (the logarithms will not be avoided anyway, as the other commenter said) and for intuition-gaining reasons. Multiplication of two functions is more intuitively understood (as one can work stuff such as signs, monotonicity etc easily) than exponentiation involving one function in the base and one in the exponent.
Plus this approach (IMO) visualised better than the proposed alternative.
I think it's arguable that OP's approach is cleaner for some value of clean, but the article's approach gave me happy flashbacks to doing number theory in undergrad that OP's approach didn't.
When communicating to the wider world, aesthetics do matter, and yeah, everything you said as well.
