Why is it advised to control the Type I error rate (e.g. Turkey's HSD) when conducting several pairwise comparisons, but not when assessing the significance of several coefficient estimates when conducting say OLS regression?
If your goal is confirmation through hypothesis tests, you should correct for the FWER (or FDR), regardless of the type of model used. If you have a source for the claim to the contrary, please include it in your question.
However, confirmation isn't the only reason someone would use linear regression. You may want to simply predict the outcome variable, or you might just be interested in the magnitude of the effects that the explanatory variables have on the outcome. Personally, I am rarely interested in the $p$-values of my linear models.
Even if you are interested in the $p$-values of a linear regression, which $p$-values you should correct for multiple testing depends on what you are doing, for example:
- Whether the intercept differs significantly from $0$ is rarely interesting. Including this $p$-value in the correction can inflate the type II error rate by increasing the number of tests, or even increase the type I error rate by including a nonsense significant result (in case of FDR correction);
- If your research question revolves around the effect of a single explanatory variable, but you want to include potential confounders, there is no need to even look at those other variables' $p$-values;
- Similarly, if your research questions concerns the presence of a (significant) interaction effect, the significance of the marginal effects may be irrelevant.
For this reason, there is no standard multiple testing correction applied to most of the default summaries of linear models, but you can of course apply your own after deciding which $p$-values matter.
Contrast this with Tukey's honest significant difference: You are comparing every group with every group. Not only is this the maximum number of hypothesis tests you can perform—increasing the risk of poor inference without some standard correction applied—but it also exists exclusively to perform comparisons, whereas linear regression in general can be used for all kinds of purposes.