"They would do this by causing their star to either radiate or evaporate in just one direction, which would propel the star, along with all of its planets, to a new location in the galaxy."
But, that would propel just the star, not the planets. The planets would have to tag along because of being attracted by the star's gravity. But, that would seriously disrupt the orbits of those planets! It would be the same effect on their orbits as if they were all accelerated in one direction at the same time while the star stood still.
Maybe by using only extremely slow acceleration rates for the star, so that the planets could keep getting back into sync and not develop too much eccentricity in their orbits?
But then the "trip" would take far longer than if a space craft accelerated just itself.
Now, if you want to imagine "tractor beams" with the power to pull stars around, then those could also bring along planets with ease, even those bigger than Jupiter, by using the beams on the planets as well as the star.
(Who thinks up this stuff?)