It is actually used quite often (in fact, Bonferroni and Bonferroni-Holm can be shown to be valid tests using the closed testing principle - see below). Part of the reason why some simple procedures like Bonferroni are still so popular is of course, that they are very easy to implement and it is easy to communicate what you did.
E.g. even for the simple Bonferroni or Bonferroni-Holm writing down the tests for the intersection null hypotheses is a bit tedious. That's why the graphical approach to constructing closed testing procedures is very popular - at least in the clinical trials setting (where there is usually a trained statistician at hand that can help pre-specify a tailored testing procedure for a specific trial). This graphical approach provides a really easy way to construct valid testing procedures and makes communication easier. Various variants, extensions and improvements such as exploiting correlations, prioritization of certain hypotheses depending on which hypotheses are already rejected etc. have been proposed over the years.
Bonferroni vs. Bonferroni-Holm as closed testing procedures example
The Bonferroni test can be easily written down in terms of the closed testing principle, e.g. in the case of two null hypotheses as
- Test for $H_1 \cap H_2$: $p_1 \leq \alpha/2$ and $p_2 \leq \alpha/2$
- Test for elementary null hypothesis $H_1$: $p_1 \leq \alpha/2$
- Test for elementary null hypothesis $H_2$: $p_2 \leq \alpha/2$
with $H_1$ rejected overall, if both $H_1$ and $H_1 \cap H_2$ are rejected. Thus, we know it is a valid test. You also immediately see that for the elementary null hypotheses $H_1$ and $H_2$ you are using a test that is not exhausting the significance level and that the Bonferroni-Holm test
- Test for $H_1$ and $H_2$: $p_1 \leq \alpha/2$ and $p_2 \leq \alpha/2$
- Test for $H_1$: $p_1 \leq \alpha$
- Test for $H_2$: $p_2 \leq \alpha$
is uniformly more powerful.
Figure 1: Illustration of the graphical approach for displaying these two procedures
Power considerations and other considerations
Usually, there's no downside in terms of power to using a closed testing procedure - if anything it tends to be easier to make sure the test fully exhausts the significance level. It can sometimes be a bit tricky to build in some desired features (e.g. exploiting a known correlation) while keep some other desired contraints (e.g. that the closed testing procedure can be easily displayed graphically due to the consistency requirement on the testing procedure).