Simulating the OLS Consistency When X and ε are Dependent

This morning I saw an interesting tweet by Nick HK, who ran a simulation to convince himself of the fact that OLS consistency only requires Corr(X, ε) = 0 but not independence:

I still find it hard to believe that OLS consistency only relies on X being uncorrelated with ε, not independent of ε, and I occasionally run a simulation to re-convince myself.

I had no doubt …