It occurs to me that you could dislodge 10,000 comets and send them toward the sun and the likelihood of a direct impact on Earth would still be statistically unlikely. Thus, for the "every 27,000,000 years" theory to hold up mathematically, a truly dizzying number of objects would have to be sent in.
That’s why the idea leads to the need for another star to get close enough to the Kuiper Belt to knock enough comets loose to ensure a strike. From what I read, the astronomers stated that nothing else could do it, and that it is possible we failed to observe that star.