Firstly, causality is not the result of a model. Causal estimates are a combination of statistics and assumptions. A model is simply fitting a line to your data and summarizing the observed joint distribution. Line fitting is the realm of predictive inference which is simply trying to find the associations in the data. Causal inference, however, is working with the observed distribution to impute an unobserved distribution and then take the difference between the two (roughly).
In your case, if we look at it from the predictive inference perspective, you are trying to understand the association between two variables (assume binary). You could do this with 2x2 tables, but then you think that variable $Z$ would change the observed distribution of this 2x2 table so you segment and run again. You could do this kind of bivariate and multivariate analysis all day to draw a picture of the associations. Eventually, though, the associations become too complex and you decide to run a model which will summarize all of these tables for you. From what I understand, this is where you are at.
Now you're making your summary associational model, but you've become concerned about omitted variable bias. This is something I see often when people blur causal inference and predictive inference. If you are doing predictive inference, the fact is OVB doesn't matter. The reason it doesn't matter is that you aren't making the assumptions required to ever actually rule it out. Long story short, causal inference uses assumptions that allow for the ruling out of any OVB. You will never achieve that with a model alone.
Therefore, the decision to do pooled OLS or fixed effects (I'm assuming on year?) is different. Pooling would be making the assumption that the joint distribution really doesn't change from year to year so it's better to increase your sample size and fit the line. Fixed effects, however, says that, yes, time does matter so I want to subgroup each year and then roll it all up into the single coefficient. Think line fitting, not OVB when doing predictive inference. If you are doing causal inference, however, you need to step back and think identification strategy before estimation strategy.