I've been wondering about these things myself for a long time too.
I have only ever read about the gravity-time relationship as a simple assertion, and I suspect that's all anyone else knows about it too. That is, if you had the absolute, true answer to this question you would know something that no other physicist on Earth knows right now. I don't think scientists would be shy in admitting if this is the case.
That said, I've had some impressions (don't confuse my opinions with facts!):
Maybe time dilation causes what we see as gravity and not the other way around. This is at least half true. Light bends in a gravitational field in part due to bent space, but also because the speed of light is slower the deeper you are in a gravity well. As a result, light is 'refracted' or bent around objects like the Sun (in part) because the wave front propagates more slowly on one side than the other. Before the 1919 solar eclipse measurements, Einstein got his math wrong by only expecting time dilation to bend the starlight. Later he corrected this by also including space curvature, which as far as I understand doubled the amount of bending expected. So half of the 'falling' of light is really time dilation.
Anyway if you chose to model all matter as little knots of light, then this same refraction (my term) would cause all matter to drift inward to wherever time is slowest. This does nothing to answer your question, but it does turn it inside out!
Another, contradictory possibility is that time dilation is caused by space being more timelike and vice versa near a large mass. That is, visualize the classic funnel of curved space we always see in magazines and note that its surface is more and more slanted near the center. If you assume the future is 'up' in this model, then for someone close to the center, their progression in time will be at more acute of an angle to their space, so what would have been time progression instead becomes more spatial acceleration and less time.
I may be completely wrong but I have been reading and thinking about these same questions for decades and this is the best I can offer.
BTW FTLdrummer is a great handle. FTLguitar would be thrash?