Endogenous variable problem or why to use 2SLS? Consider linear model $$y = \beta_0 + \beta_1x_1 + \dots + \beta_{k-1}x_{k-1} + \beta_kx_k + u $$ where $x_k$ is correlated with $u$ which results into inconsistent estimators of $\beta$.
As I understand endogenouity of $x_k$ arrises from the fact that it depends on some exogenous variables which are captured in error term $u$. One possible solution is to use 2SLS procedure which requires to find such vector $z$ that $\mathbb{E}(z'u) = 0$ and $Cov(z, x_k) \neq 0$.
My question is somewhat naive but I don't quite understand why one could not just add this additional exogenous variables that are captured in $z$ into initial equation?
$$y = \gamma_0 + \gamma_1x_1 + \dots + \gamma_{k-1}x_{k-1} + \gamma_kx_k + \theta_1z_1 + \dots + \theta_lz_l +e $$
Now $x_k$ is not correlated with error term and estimator for $\gamma_k$ should be consistent but it probably not true or why to invent such procedure like 2SLS. What do I miss?
 A: Here's a fairly high-level explanation. Others might chime in with more specifics. 
If you enter the exogenous variable $z$ into the initial linear equation, you are finding the unique variation in $x_k$, net of $z$ etc. But what are you interested in for IV and 2SLS is the variation in $x_k$ that is associated with $z$, i.e. the variation that is exogenous to the potential omitted variables captured by $e$. By assumption, if you have a good instrument, corr($z_i$,$e_i$)==0. So, $z$ is not going to soak up any variation captured in $e$. The first stage regression (ideally) finds the variation is $x_k$ that is not associated with $e$ and thus not subject to OVB. 
Think about this in terms of a coin flip that randomly assigns to people to a group. Imagine putting a dummy for assignment coin flip into an OLS regression. What does the $\beta$ of interest then tell you? The impact of $x_k$ that is independent of random assignment. That's not what you want to know. You want to know what the impact of the variable of interest that is due to random chance (i.e. assignment by coin flip).  
And, BTW, in your example, $x_k$ is still potentially correlated with the error term, even if you include $z$! The instrument does not account for all the variation in the error term. The first stage gets that variation in $x_k$ that is associated with $z$, so that the fitted values in the second stage regression are therefore not correlated with $e$.
