I'm confused - he's addressing the point made in the article by Ed Pegg that things are interseting if you allow log and sqrt. Surely that means he's allowed to use log and sqrt, and in particular, to use them in rational expressions.
The point of the exercise is to find a short approximation of pi, no? If you allow the use of e, then you can define pi. What he wrote is not an approximation, it is pi. Surely at that point you've defeated the point of the exercise.
Once you allow sqrt and ln (or sqrt, log, and e) the problem is silly. He explicitly allows, see the bottom of the article, sqrt, log, and irrational numbers.
Once he introduces the square root he introduces imaginary numbers, sqrt(-1), and transcendental numbers, for example the Gelfond–Schneider constant 2^sqrt(2).
Now we're really down the rabbit hole of someone else's intent, but personally, I assumed he was still trying to maintain some restriction. So, no imaginary numbers, no trascendental numbers. It's easy to restrict what we take the root of, and what we do with the potential irrational result of such roots, to ensure that. As you pointed out, to not do so defeats the purpose of the exercise, and I think my assumption is both reasonable and charitable.
So I don't really understand what your point is.