Is the core specifically the cause of gravity in either model? In RE, it is the entire mass of the Earth which causes gravity, not merely the core.
No, its not the core, it's the mass of an object which determines it's gravitational pull. Note that I said mass, no size- a small, densely packed object, like a dead white dwarf star, has much greater gravitational pull than Jupiter, which is many times the size of a white dwarf star.
Which is why the 'stratellites' which 'orbit' above a flat earth are so ludicrous, which you've been told 100 times before so I'll save your blood pressure and not going into it again.