The standard error of the variable of interest $x$ can be calculated as
As usual, $\sigma_\varepsilon^2=\sum_i\varepsilon_i^2 $ is the variance of the regression error and $VIF_x$ is the variance inflation factor of $x$.
If one now controls for a second variable $z$ (which happens to be highly significant), one inevitably reduces the residuals $\varepsilon_i$ what leads in turn to a reduction of $\sigma_\varepsilon^2$. Because $z$ is not a confounder, $VIF_x$ remains unchanged. Ultimately, $s.e.(\hat\beta_x)$ would go down.
If my thinking is right, then one would try to control in large datasets (say, 100k observations) for as many highly significant control variables as possible. That is because the loss in degrees of freedom is negligible and the $p$-value of the variable of interest goes down.
Whether to control for non-confounders seems to be quite an important question to get right in applied statistics. I am therefore wondering whether my argument is correct or whether I got something wrong?
Best wishes, Tom