How does this cause a point particle to accelerate towards the sun? Must be something about the gradient, but how does the gradient of time cause you to curve towards the sun?
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.
I disagree, an actual point particle with a mass should have an event horizon. Using terms without baggage helps avoid such misleading assumptions.
“Particle” isn’t great but it beats “point particle”
The mathematics of this is a bit too complex to reproduce in a comment here, but in, say, the Earth's gravitational field, taking this effect into account (approximating GR as a field of locally varying clocks, then allowing, e.g., an electron's wavefunction to evolve on that spacetime) would reproduce gravitational acceleration / free fall towards the Earth.
Said differently: this is precisely the kind of nuanced scenario where getting sloppy with metaphors gets you into trouble very quickly. Quantum mechanics in curved spacetime is not to be dabbled with lightly.