Apparently, the chord of contact on a hyperbola from any point on the asymptote is parallel to the asymptote itself.
I've reasoned this out by showing that whatever slope one of the tangents from the point has, the second tangent is the asymptote itself. It feels like the chord of contact would therefore tend towards the slope of the asymptote, as the meeting point of that second tangent with the hyperbola lies at an infinite distance.
Is this reasoning correct?
Your reasoning is sound, but not quite rigorous yet. Try to add some math to it. Meanwhile, here's another procedure:
The chord of contact on a hyperbola from an external point is given by $T=0$. Hence the chord on $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ from some point $(x_1,\frac {b x_1}{a})$ is given by:
$$\frac {x x_1}{a^2}-\frac {yx_1}{ab}=1$$ Thus the slope of this chord is $\frac ba$, and hence parallel to the asymptote.