Pros for Death Star Theory
explains the periodicity of impacts on Earth.
explains the source of the projectiles.
Cons for Death Star Theory
a 26 million year orbit for the planet/star body would take it great distances from our Sun, to the extent where the body's orbit would be perturbed by nearby stars. Thus, affecting its periodicity.
no evidence supports the presence of such a planet.
data used has numerous problems. For example, the data fits only for the last 3 extinctions (K/T, late Eocene and mid-Miocene). The fit become highly problematic once you try to fit the extinctions before the K/T one.
data is only for the last 250 Ma.
their models in assigning families and genera to a geological period were incorrect. They assumed at all extinctions occurred simultaneously everywhere on the globe, exactly at the end of the a geological period.