Does endogeneity matter when neither independent variable nor error term are correlated with dependent variable? if the double arrows show that X and the error term are correlated, but that neither variable affects Y, is endogeneity a problem in this scenario? Why or why not?
Assume the simplest linear regression model

$$y = bx + u$$

The OLS etimator for $b$ is

$$\hat b_{OLS} = \frac {\sum x_iy_i}{\sum x_i^2} = b + \frac {\sum x_ie_i}{\sum x_i^2}$$

Whatever the true $b$ is (it is zero in your case, as another answer pointed out) , the fact is that

$$\text{plim} \frac {n^{-1}\sum x_ie_i}{n^{-1}\sum x_i^2} \neq 0 $$

and so the OLS estimator will be inconsitent (and also, biased too).