Gravity should be uniform regardless of height on an infinite plane
Incorrect.
please elaborate
A very curious thing about the existence of an infinite, completely flat plane is that, whatever the height you get from the plane, the gravitational pull will be the same. Of course, if you want an infinite plane you have to start with the magical appearance of an infinite amount of matter, just flat lying around you and you have not even a clue as to why it shaped itself in an infinite plane, but mathematics permit us some understanding of impossible scenarios.
This scenario is complicated by the existence of mountains, which have their own gravitational pull and could start the self-destruction of the infinite plane, and with tectonic plates that, in real life, have moved through the South Pole, something that is impossible on a real Earth. And to complicate things even more, Tom Bishop decided that the stars have gravitational pull over us, so whatever measurement of gravitational pull you get on whatever place on Earth, the Flat Earthers will say they predicted that result.
In reality, the idea of an infinite plane with gravity is as bad as the flat Earth with Universal Acceleration, except for the explanation of why we are not being burned by the rocket under us that accelerates us at 9.8 m/s/s for billions of years on end.