> These numbers are wildly exaggerated, but isn't this also the wrong "direction" of relativistic time dilation? In the frame of the planet orbiting the hypervelocity star, the rest of the universe is moving at 0.5c, so the planet would measure all clocks in the rest of the universe to be running slower by a factor of 1/sqrt(1-0.5^2) = 1.15. Astronomers on this planet would measure stellar lifecycles (and every other physical process) to proceed slower, not faster.
You have it backwards (easy to do with such a thought experiment!). The direction of time dialation originally stated was correct. Clocks on a body moving at ludicrous speeds will be slower (to the observers present, who are able to look at those clocks) than clocks on a body not moving at ludicrous speeds. Which is to say, if you did a loop around the galaxy at light speed (hypothetical, of course) and came back to Earth and compared your clock with a terrestrial clock, your clock would be waaaay behind, and you would be younger than the people you left behind. This effect is measurable and provable even in the orbit of our own planet (albeit, at a miniscule scale). And so while your time did in fact move more slowly, the overall effect is that you time travelled into the future because everyone else was moving much faster. This of course assumes that you are able to escape intact, which is most unlikely (again, this is just a thought experiment, for argument's sake).
In your defense, I know how easy it is to get turned around in such a thought experiment. Theoretically speaking from the perspective of the par-luminal traveller, it's easy to think "my clock moves slower, so am I slower?" In a way, yes, you are! Your reality moves slower relative to theirs, and for those not travelling at light speed time passes more quickly relative to your time. Objectively speaking, a human lifecycle on a very fast moving planet will equate to many, many lifecycles on a slow moving planet, if all other things are equal. And so while you are truly travelling much faster, your time moves slower, relatively speaking.
Remember as a kid being told about a hypothetical traveller that enters a black hole, and at one point they become a long spaghetti string and then simply freeze in time? That's because they are moving much faster than us, and from the outside looking in they appear to be frozen. The light emitted from their mass is but an echo. They have moved far beyond the light that we see. From the inside looking out, the universe would move much faster.
Again, this is an extreme example used for clarification, but it might not be as extreme as you think. Consider a solar system that gets caught within the grasp of a black hole. Initially the tidal forces tear it apart, but over eons that mass might begin acting like a normal solar system again, and harbor life once more.