A powerful test for any distribution Simulations by Razali et al (2011) showed that the Shapiro-Wilk test of normality provided the most power (at a fixed significance level) when compared to Anderson-Darling, Kolmogorov-Smirnov, etc. My question is this:
For data $X_1, \cdots, X_n$, suppose I want to test 
\begin{align*}
H_0: X_i \sim F \qquad \text{vs} \qquad H_1: X_i \not\sim  F
\end{align*} 
for some continuous distribution $F$. This is equivalent to 
\begin{align*}
H_0: \Phi^{-1}(F(X_i)) \sim N(0, 1) \qquad \text{vs} \qquad H_1: \Phi^{-1}(F(X_i)) \not\sim N(0, 1)
\end{align*} 
through probability integral transforms, where $\Phi$ is the standard normal CDF. Since Shapiro-Wilk does well in terms of power, this procedure would give us a rather powerful test. Why isn't this idea more prevalent in the literature?

Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests"

 A: The Shapiro-Wilk test is for testing normality with unspecified $\mu$ and $\sigma$.
In proposing to transform to uniformity and then standard normal, you're assmuing that $F$ is completely specified (no unknown parameters), so that $F(X)$ is standard uniform; and then that transformation to normality will produce only a standard normal (fully specified distribution). 
Consequently:


*

*If $F$ is completely known, the Shapiro-Wilk test will be wasting power on treating-as-unspecified parameters in the normal that are actually known; it may be better in that case to use a test specifically for a fully specified distribution such as the Anderson-Darling. (It might still do quite well, but I wouldn't automatically expect it would do quite as well as in the situation it was designed for and which the power comparison you mention was based on.)

*If some parameters of $F$ are unknown, estimating those parameters of $F$ and computing $\hat{F}(X_i)$ will produce a conservative (and therefore lower-power than if it were completely specified) test. (This is akin to using a Kolmogorov-Smirnov test when you should be using a Lilliefors test, and with similar lowering of the power.)

*The sorts of alternatives you may be interested in for $F$ may not be "typical" alternatives that the Shapiro-Wilk does particularly well on. There are alternatives where it's not best or even second best and it may be that for some choices of $F$ that the interesting alternatives might be focused on what would be a narrow subset of cases at the normal (e.g. perhaps this might be the case when testing the Pareto, where in some applications extreme-upper-tail differences might be of substantial interest, but would correspond to only a small subset of typical alternatives of interest when looking at testing normality)
