Linear regression, good and bad controls, omitted variable error, and causal graphs This is my first post on this site, and I would really like to thank everyone who engages in this community. I have learned a lot from reading both the questions and answers. My questions are at the bottom of this post, but I provide some motivation first, hoping to make clear where they come from and what has been my thinking so far.
I’m trying to reconcile what I’ve learned from the literature on causal graphs with my knowledge from textbook econometrics which is mostly based on the excellent “Introductory Econometrics: A modern approach” by Jeffrey Wooldridge.
In his textbook Wooldridge introduces in “Chapter 2: The simple linear regression model” the model $y=b0+b1*x+u$ where $b1$ “does measure the effect of $x$, holding all other factors (in $u$) fixed” and $u$ is the error term including “all factors affecting $y$ other than $x$”. He then explains that we can only interpret our estimator of $b1$, i.e., $\hat{b1}$, causally (as ceteris paribus effect), if the zero conditional mean assumption $E(u|x)=E(u)=0$ holds. The relevant part of this assumption is that the error term $u$ is mean independent of $x$, i.e., $E(u|x)=E(u)$. Most often, we think of this in terms of the implication that $x$ and $u$ should not be correlated, i.e., $cov(x, u)=0$.
The classical example concerns the effect of education on wages. The respective model is $wage=b0+b1*education + u$. Here it is argued that the error term $u$ includes the innate ability. Because individuals with a higher innate ability will have higher education (due to innate ability causing education), the error term $u$ and education will be correlated, meaning that the zero conditional mean assumption is violated. In other words, we have an omitted variable error, and $\hat{b1}$ cannot be interpreted causally (as ceteris paribus effect). The solution (I know there are different ones, but that’s not my point here) would be to measure innate ability (leaving discussions of whether this is possible aside), moving it from the error term $u$ into our model by making it an additional $x$ variable. In this case, the innate ability would be a "good control variable".
Later in the chapter “6-3c Controlling for Too Many Factors in Regression Analysis”, Wooldridge discusses another example where interest is in the causal effect of a beer tax on fatalities. He then explains that we should not include beer consumption in our respective model, because the effect of a beer tax on fatalities will be mostly due to beer taxes reducing beer consumption and the latter is causing fatalities. In this case, beer consumption "would be a bad control variable". I, of course, agree with this explanation and I understand it intuitively as well as when I consider what I’ve learned from the literature on causal graphs (i.e., adjusting for beer consumption would violate the backdoor criterion).
But I’m wondering how Wooldridge would formally justify his very clear statement “Are we committing an omitted variables error? The answer is no” (as he does not refer to criteria such as the backdoor criterion as a justification). Consider the model $fatalities = b0 + b1*beertax + u$. If we think about it the same way as for the model relating wages to education, I would think that beer consumption is included in the error term $u$ as it is among all factors affecting fatalities other than beer tax. And because beer tax and beer consumption are correlated (due to beer tax causing beer consumption), the error term $u$ and beer tax will be correlated and, therefore, the error term $u$ will not be mean independent of beer tax violating the assumption. In other words, we have an omitted variable error.
So, my question is how to reconcile this. Are we formally really not committing an omitted variable error, and, therefore not violating the mean independence assumption? If so why is this case and where am I wrong?
I can think of two ways to reconcile it myself, but I’m not really sure about either:
First, we are not committing an omitted variable error in the second example, because beer consumption is, against my current thinking, not part of the error term $u$. Then the error term $u$ would not be correlated with beer tax and everything would be ok. But then my question really would be, why beer consumption, which from my understanding does have an independent effect on fatalities, would not be part of the error term $u$. Would it be because all the variation in beer consumption that is caused by beer tax does not represent an independent effect on fatalities and the remaining variation in beer consumption (coming from any other sources of beer consumption and representing an independent effect) is indeed mean independent of beer tax? If so, how would we show this formally using the notation of Wooldridge?
Second, we are committing an omitted variable error, but the mean independence assumption as presented in Wooldridge formally refers to the direct causal effect rather than the total causal effect, so it would be related to the single door criterion instead of the backdoor criterion in the language of causal graphs. This seems unlikely to me as Wooldridge is really clear in stating that we are not committing an omitted variable error and therefore the mean independence assumption should not be violated.
I hope my questions are articulated clearly enough, but let me know if I should edit certain parts. Thank you for your answers!
Edit based on Michael's answer:
Thanks, Michael. Your answer below is already very helpful. Allow me to follow up on it, just to make sure I understand you correctly.
Wooldridge defines the error term $u$ as "The variable $u$, called the error term or disturbance in the relationship, represents factors other than $x$ that affect $y$." when discussing the simple linear regression model and as "Just as in simple regression, the variable $u$ is the error term or disturbance. It contains factors other than $x_1, x_2, ..., x_k$ that affect $y$." when discussing multiple regression. To me, this definition is the same as what you state to be not quite the interpretation of the error term. So, because this definition of the error term is not precise, I wrongly assumed that beer consumption would be included in the error term, which, in fact, it is not. To show this you pointed me to the "chain rule". Let me try to illustrate your argument in detail. We can think of the example concerning beer taxes ($x_1$), beer consumption ($x_2$), and fatalities ($y$) as a system of linear structural equations:
$x_1 = e_{x_1}$
$x_2 = a_0 + a_1*x_1+e_{x_2}$
$y = g_0 + g_1*x_1+g_2*x_2+e_y$
Now I can rewrite the equation for $y$ by inserting the equation for $x_2$
$y = g_0 + g_1*x_1+g_2*(a_0 + a_1*x_1+e_{x_2})+e_y$
Rearranging this equation gives
$y = (g_0 + g_2*a_0) + (g_1+a_1*g_2)*x_1 + (g_2*e_{x_2}+e_y)$
Redefining the terms $(g_0+g_2*a_0)=b_0$, $(g_1+a_1*g_2)=b_1$, and $(g_2*e_{x_2}+e_y)=u$ gives us the familiar looking model
$y = b_0 + b_1*x_1 + u$
Here it can be seen that $u$ does not include $x_2$ (only its independent part $e_{x_2}$) and that $x_1$ is not correlated with $u$, as it is not correlated with any of the components of $u$.
In contrast, if we think of the example concerning education, innate ability, and wages as a system of linear structural equations the same "trick" of chaining equations will not be possible and ultimately, $x_2$ (innate ability), will be part of the error term after having redefined the terms.
So, after all, the first "solution" in my original post would be right?

First, we are not committing an omitted variable error in the second example, because beer consumption is ... not part of the error term $u$.

If what I have written here is correct, the question is what a more precise definition of the error term would be. Should it be something like this?
"The variable $u$, called the error term or disturbance in the relationship, represents factors other than $x$ that affect $y$ and that themselves are not affected by $x$?"
With such a definition, I would not have made the mistake to think that beer consumption is part of the error term.
Edit
In a comment below Michael suggest rephrasing the more precise definition to "The variable $u$, called the error term or disturbance in the relationship, represents factors other than $x$ that affect $y$ and that themselves are not channels of the effect of $x$ on $y$?"
 A: 
But then my question really would be, why beer consumption,
which...does have an...effect on fatalities, would not be part of the
error term u?

The error term in the linear model is not interpreted to contain quite "...all factors affecting fatalities other than beer tax" (although you can see why it's not unreasonable for make this initial claim for pedagogical reasons).
If the regressor $x$ channels its effect on $y$ through $z$, and that is the only way $z$ influences $y$, you can see why the model
$$
y = \beta_0 + \beta_1 x + \beta_2 z + u
$$
would not make sense---the partial effects of $x$ and $z$ on $y$ are not additive. Rather, the functional form that incorporates both $x$ and $z$ would be a type of composition (e.g. $y(z(x))$) and the partial effects operate via a chain rule. Therefore adding $z$ would be adding bad control. That is Wooldrige's point in the example where $x =$ beer tax and $z= $beer consumption. Empirically, some of the partial effect of $x$ would be mistakenly captured by the coefficient on $z$, whose effect on $y$ derives solely from that of $x$.
Notice this is not the case for your example of omitted variable bias,
$$
wage=\beta_0 + \beta_1*education + u.
$$
The effect of education on wage is not channeled through the omitted variable. It is reasonable to assume the partial effects of education and omitted variable are additive.
Another situation where a variable that has an effect on $y$ but is not contained in $u$ is simultaneous equations. The classic example is demand function estimation. Consider the demand-supply system
\begin{align*}
q &= \beta_{0,d} + \beta_{1,d} p + \epsilon_d \\
q &= \beta_{0,s} + \beta_{1,s} p + \epsilon_s.
\end{align*}
The supply shock---error term $\epsilon_s$ in the supply equation---has an effect on quantity $q$ but is not contained in the demand shock $\epsilon_d$. In this case, this can be exploited---an instrumental variable can be extracted from the supply shock to instrument $p$ in the demand equation.

...the backdoor criterion...

I believe neither the back-door nor front-door criterion describes the case of bad control.
The backdoor adjustment corresponds to adding controls for omitted variable bias, while the front-door adjustment covers IV as a special case.
(Other folks more familiar with do-calculus would be able to comment more precisely on this.)
