Why use normal approximation?
It's as simple as saying that it's always better to use more information than less. The equation (1) uses Chebyshev's theorem. Note, how it doesn't use any information about your distribution's shape, i.e. it works for any distribution with a given variance. Hence, if you use some information about your distribution's shape you must get a better approximation. If you knew that your distribution is Gaussian, then by using this knowledge you get a better estimate.
Since, you're already applying the central limit theorem, why not use the Gaussian approximation of the bounds? They're going to be better, actually, tighter (or sharper) because these estimates are based on the knowledge of the shape which is an additional piece of information.
The rule of thumb 30 is a myth, which benefits from the confirmation bias. It just keeps being copied from one book to another. Once I found a reference suggesting this rule in a paper in 1950s. It wasn't any kind of solid proof, as I recall. It was some sort of empirical study. Basically, the only reason it's used is because it sort of works. You don't see it violated badly often.
UPDATE Look up the paper by Zachary R. Smith and Craig S. Wells "Central Limit Theorem and Sample Size". They present an empirical study of the convergence to CLT for different kinds of distributions. The magic number 30 doesn't work in many cases, of course.