Confused by extremely high autocorrelation
Most of the values of the random walk $x$ are large ($\omega(1)$) with high probability. Therefore the difference between $x_i$ and $x_{i+1}$, which is $\pm 1$, is small. So the random walk correlates well with itself.

You can get results even closer to $1$ if you start your random walk from some position far away from the origin. Try it out!