I've read [Zero conditional mean assumption (how can in not hold?)](https://stats.stackexchange.com/questions/210083/zero-conditional-mean-assumption-how-can-in-not-hold). 

In linear regression, we assume the model follows
$$y\_i = \beta\_0 + \beta\_1 x\_i + \epsilon\_i
$$
under the assumption that $\mathbb{E}(\epsilon|X) = 0$. 

However, I don't see why we must have that assumption, even if $\epsilon\_i$ and $x\_i$ are correlated. 

Because if $\mathbb{E}(\epsilon|X) = c \neq 0$, then why can't we just assume another model
$$y\_i = \underline{\beta}\_0 + \beta\_1 x\_i + \underline{\epsilon}\_i$$
where $\underline{\beta}\_0 = \beta\_0 + c$ and $\underline{\epsilon}\_i = \epsilon\_i - c$ 

and applying the typical algorithm, we can find estimates for these new parameters?