I have not been able to find satisfactory explanations for the logic behind Tukey's HSD test. However, the resources I have been able to find all concern it's use. It's supposed to make multiple pairwise comparisons of means and identify those pairs which are significantly not equal.

Everything seems to hinge on the Studentized range distribution. Wikipedia says:

Suppose that we take a sample of size n from each of k populations with the same [emphasis mine] normal distribution $N(\mu,\sigma^2)$ and suppose that $y_{\min}$ is the smallest of these sample means and $y_{\max}$ is the largest of these sample means, and suppose $s^2$ is the pooled sample variance from these samples. Then the following statistic has a Studentized range distribution.

$$ q=\frac{{\overline {y}}_{\max }-{\overline {y}}_{\min }}{s/\sqrt {n}} $$

Accepting this, it would mean that assuming that all groups have identical true variance and mean, we should expect all pairwise mean differences to be smaller (in absolute value) than a certain value (which depends on our desired type 1 error significance level). However, my conclusion from this is that, in practice, if I observe that two groups have significantly different means, then the assumption that all $k$ populations have the same mean must be false. I struggle to understand how one can conclude that that specific pair of means is different?

It would make more sense to me if in the result from Wikipedia, the assumptions were relaxed so that each group could have its own mean, and the result still held.