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.

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 limited sample sizes and are just looking for some publishable *p*-value, as many people are, this makes it attractive.

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