I few thoughts that came to me as I read this article:
1. Any collision of a body large enough to take the Moon completely out of Earth orbit would also have a profound effect on the orbit of the Earth around the Sun, even without touching the Earth. In the sense of momentum transfer by gravitational attraction, rather than physical contact, it would still be similar to a collision of two billiard balls. Earth would get a substantial push in some direction, substantially changing its orbit around the Sun.
2. But, a lot of this story has to do with not having what the Moon normally does to the Earth, rather than the effects of something that could move it away. In that regard, the story seems to neglect the effects of the Moon's gravity on the "tides" of things other than the oceans. It isn't just the water surface that rises and falls due to the Moon's gravity. The Earth's crust also rises and falls. Not as much as Jupiter's gravity apparently affects Io, but I wonder how much it affects volcanism on Earth. And, does the Moon cause the plate tectonic motions here on Earth? Directly or indirectly? We aren't seeing similar motions on other small planets without large moons.
And, while we wonder about situations other than what we really have: I wonder what would have happened if the postulated collision between Theia and proto-Earth had resulted in 2 very similar planets of about the same size and composition as Earth. Instead of going to a lifeless orb of rock and (maybe) frozen water, could we instead have the opportunity to go to another habitable world? Or, maybe its slightly superior denizens might be coming to our world first. At least it would give us a population of 2 planets where life evolved to compare and contrast - might answer some questions we still have about such things.