I don't buy it. Even allowing counting iron as separate from what rocks can be composed of (and using mass instead of volume) you still have 30.1%+15.1%=45.2% of the Earth as oxygen and silicon (which are most certainly part of what makes a rock) at which point you've already disproved the claim Earth is more a ball of iron than a ball of rock.
A ball of iron covered with a ball of rocks is a more fair statement though, and I'd agree with that. It's just that center ball isn't most of what makes up the Earth (by any measure).
Everything up to and including the mantle is either iron or has a lot of iron. But to your point the mantle also has a lot of silica. So I guess it depends on your definition of "mostly".
Mass is the defining characteristic of a quantity of matter. Given that much of the iron is under far higher compression than the outer layers of silicate rock, this also advantages iron.
By mass, iron (32.1%) is still a minority constituent of the Earth.
A ball of iron covered with a ball of rocks is a more fair statement though, and I'd agree with that. It's just that center ball isn't most of what makes up the Earth (by any measure).