The potato radius is derived from Newtonian gravity, as both of your links demonstrate.
Here’s the point dude: you don’t have the training or knowledge to understand what the field equations say or do, so just don’t pretend otherwise. Because I’ll call you on it.
It’s important that both FEers and RErs alike discuss evidence correctly.
The field equations say nothing about the shapes of planets. Now, as you suggested, let’s move on. You wish to revise #1, so do that, and let’s see if we can trace a path through your numbered arguments that’s cogent.