Indeed an omnibus test is not needed and some multiple inference procedures like Bonferroni or Bonferroni-Holm are not limited to an ANOVA/mean comparison setting. They are often discussed in the context of post-hoc tests in textbooks or associated with ANOVA in statistical software but if you look up papers on the topic (e.g. Holm, 1979), you will find out that they were originally discussed in a much broader setting and you certainly can “skip the ANOVA” if you wish.

One reason people still run ANOVAs is that pairwise comparisons with something like a Bonferroni adjustment have lower power (sometimes much lower). Tukey HSD and the omnibus test can have higher power and even if the pairwise comparisons do not reveal anything, the ANOVA F-test is already a result. If you work with small and haphazardly defined samples and are just looking for some publishable *p*-value, as many people are, this makes it attractive even if you always intended to do pairwise comparisons as well. Also, if you really care about any possible difference (as opposed to specific pairwise comparisons or knowing which means differ), then the ANOVA omnibus test is really the test you want.

Finally, many people are happy with this routine (ANOVA followed by post-hoc tests) and just don't know that the Bonferroni inequalities are very general results that have nothing to do with ANOVA, that you can also run more focused planned comparisons or do a whole lot of things beside running tests. It's certainly not easy to realize this if you are working from some of the most popular “cookbooks” in applied disciplines and that explains many common practices (even if it does not quite *justifies* them).

Holm, S. (1979). A simple sequentially rejective multiple test procedure. *Scandinavian Journal of Statistics,* 6 (2), 65–70.