I’m no physicist, but it seems to me they made this new death star partially believable. Partially.
So, one could imagine somehow harnessing a star and draw power from that to destroy a planet. The stream of matter moving from a star to the death star’s energy collector seems to be inspired by how 2 stars would compete on each other on gravity and attract each other’s matter. Also the hypothetical Dyson sphere could be part of it, although that would probably bigger than the planet that is the new Death Star.
But what’s important here to consider is gravity, or well, the warping of space-time. Because whether you attract just new energy or actual matter from another object that wasn’t already a part of you, adds to your gravity. So as the energy intake increases, the mass and gravity would increase as well. Even if the planet could withstand the pull of star that a Dyson sphere could do, or was somehow protected from it, the people and everything else flying around it would not be protected from it.
Meaning by the time all the star’s energy was taken in, it would also have squished the people on it, and buildings, and trees, and crashed all the spaceships around it. The slowing down (or faster? errr time dilation is complicated) of time would probably be less relevant at that time since anyone caring about it would be dead. It could probably work if it was remote controlled, but that would probably be less exciting.
I can also imagine the collapse of the planet would form another star. Though I wouldn’t know if it would form that quickly as it did in the movie. But who knows, it might.
Still, nice try.