In FE, rhumb lines are always geodesics. In real life, rhumb lines are not always geodesics. Hence, the Earth is not flat.
Can anyone refute this?
You've never proven that Rhumb lines are always geodesics on FE. Common sense says that even on an FE, Rhumb lines would still not always be geodesics.
And as for your constant "direction" changing to keep going East, have you read the FAQ? It's pretty self-explanatory.