I have an initial regression of Y on X and Z. Both of my coefficients on X and Z are non-zero and strongly statistically significant. X and Z are correlated but I am told collinearity shouldn't be an overwhelming issue, OVB would be a stronger one, so I should keep both variables. I can interpret the coefficient on X as the effect of X on Y controlling for Z, and likewise can interpret the coefficient on Z as the effect of Z on Y controlling for X.
Now, at the same time, I am considering using Z as an instrument for a regression of Y on X. I have read on several posts on this forum that the fact that coefficient on Z is non-zero in the initial regression above doesn't invalidate the possibility that Z is a good instrument (the same DAG is often posted, which I have to say I do not fully understand).
For me there is a contradiction between these two assertions. Is there really a contradiction or have I misunderstood something? My worry is that as the coefficient on Z in the initial regression is non-zero and highly significant, it does reduce its validity as an instrument given I have already controlled for X in that regression, so the effect I'm estimating is in theory a direct effect of Z on Y independent of X, which seems fairly large. Anyone can help me build an intuitive argument (in words) about why the positive coefficient on Z is actually fine (if this is indeed the case)?