Why, exactly, is a unit root a problem? Suppose that we have observations $x_1,\dots,x_n$ for some process.
We want to fit an AR(k) model to these observations.
I do not understand why the naïve OLS approach to estimate our AR(k) coefficients would be inappropriate when the process has a unit root. From some simulations it seems to recover the coefficients no matter the location of the roots of the lag polynomial.
If estimating the AR coefficients is the not problem, then what is? A unit root means that uncertainty around your forecast grows with time, but that's not a problem from a mathematical point of view, it just something to keep in mind when interpreting the forecasted values.
Basically I'm asking which part of the mathematical analysis breaks down in the presence of a unit root?

The process is very different in the presence of a unit root. Many processes increase or decrease exponentially. But if there is a unit root, the process is a random walk, which is different.
