Endogeneity and the AR model?
It is not necessarily true that lagged $y$ are endogenous. That depends entirely on the structure you assume for $u_t$. If $u_t$ is white noise, there is no endogeneity, in the sense that 

$$ E(u_t|\mathbf{y}) = 0 $$

where $\mathbf{y}=\{y_{t-1}, \cdots , y_{t-p}\}$.

This is trivial to prove, once you realise that the above equality can also be expressed as 

$$E(u_t y_{t-i})=0 \,\,\quad, i=\{1, \cdots, p\} $$

using the Law of Total Expectations.

Conversely, it is trivial to show that if $u_t$ follows an autoregressive process, the above conditions does not hold anymore, in which case there is endogeneity, and OLS estimates are inconsistent.