You forgot a few things in your assumptions ...

*"The earth is an infinite plane"*

-> That means, no matter how thick, that V - Volume - is infinite

-> That means, no matter how low the density, M - Mass - is infinite

-> That means that gravity would be infinite on every possible point on this "plane"

-> But that also means that this plane **does not actually have a centre of gravity**, because all points on this plane are equally - infinitely - far away from the - technically non-existing - edge. You trying to glance over that problem by *"assuming the plane to be symmetrical via rotations"* doesn't just make it so - either infinite or a centre, not both

-> That means the distance between the centre of gravity of the two attracting masses r is undefined and can't be used in a calculation

Basically, the whole thought-experiment has already broken down at this point ... twice, on your first formula. Infinite planes can be used in math and geometry, but it doesn't work with actual physical properties, genius.

This is like one of those math "problems" created by "dividing by zero" or other tricks that make any calculation after that point mute ... "Einstein would love that!"