Why the inner core of planet Earth is not composed of the densest materials? I read on various sources the inner core of our planet is said to be mostly composed of a iron-nickel alloy, plus some lighter elements like silicon. On the other hand, I know that in general lighter elements go to the upper layers and heavier sink, because of buoyancy. So I am wondering why the densest materials are not considered in the inner core composition, namely osmium 22.59 g/cm3, iridium 22.56 g/cm3, platinum, all the way down lead.
Of course some of those heavy materials are rare. But even if they are rare, I am not sure this is the reason why they are not mentioned at all when describing the inner core.
I found a related question but not equivalent in my view because it does not address the core composition per se:
Why heavy elements don't sink to the core?
 A: 
So I am wondering why the densest materials are not considered in the inner core composition, namely osmium 22.59 g/cm3, iridium 22.56 g/cm3, platinum, all the way down lead.

Not lead. I'll address that later.
All of the elements you mentioned (including lead) are extremely rare in the universe. Iron on the other hand is the ninth most common element in the universe. Iron is formed in abundance in Type II supernovae and in exploding white dwarfs. Of the siderophile elements, only iron and nickel are common.
Iron is even more abundant in carbonaceous chondrites, which are widely thought to be the source of much of the Earth's materials. By mass, iron is the second most abundant element in carbonaceous chondrites, after oxygen. Due to planetary differentiation, iron, along with other siderophile elements, is significantly reduced in abundance in the Earth's crust compared to carbonaceous chondrites.
The other siderophile elements include (in order of abundance) nickel, manganese, cobalt, and several rare metals. Those rare metals are rare in the universe, rare in meteorites, and rare even in the center of the Earth. Manganese and cobalt together comprise a bit less than 1% of the mass of the Earth's core. All of the other siderophile elements are extremely trace elements, even in the Earth's core.
Lead is a chalcophile rather than a siderophile. Lead readily combines with sulfur to form the mineral galena. Galena is less dense than iron and it has a rather low melting point. The low melting point means that galena has a tendency to concentrate in the Earth's crust rather than its mantle. Lead is also the end point of the decay chains of uranium 238, uranium 235, and thorium 232. Uranium and thorium are highly lithophile; they are concentrated in the Earth's crust.
Lead's chemistry and the physics of radioactive decay means that lead is concentrated in the Earth's crust rather than in the Earth's core. The abundance of lead in the Earth's crust is greater than that of lead in carbonaceous chondrites.
A: Two years on and Winston´s question is still not answered.
Osmium and iridium are distinctly heavier than all other elements and their natural alloy, osmiridium, should find its way (by hydrostatics over millions of years) to the centre of the Earth.
Cox estimates the abundance of Os at $10^{-4}$ ppm and Ir at $10^{-6}$ ppm. Supposing an osmiridium alloy with a density of $22.58\;\mathrm{g}/\mathrm{cc}$, reflecting relative abundance, that means most of the terrestrial osmiridium should be in a sphere about $2\;\mathrm{km}$ in radius at the centre of the earth . . .
I suspect it is.
