The primary purpose of conducting Tukey pairwise comparisons after a significant ANOVA is to:

Determine whether the assumptions of ANOVA hold by checking residual diagnostics for each individual pairwise contrast.

Estimate the pooled variance for all future experiments without making any conclusions about differences between group means.

Identify which specific group means differ while controlling the overall Type I error rate across all pairwise tests.

Replace the need for ANOVA entirely by testing each pair sequentially without any adjustment to error rates.