really, I expect at minimum an earth sized iron core in the sun, it is probably much more.
The Earth does not have its iron core because individual iron atoms, being heavier, individually diffused downward into the core. Some heavy elements, like uranium, are actually highly concentrated in the continental crust, because chemically they partition into lower density phases.
In the Sun, with everything inside vaporized, there is and was no partitioning into distinct chemical phases of differing density. There is a chemical difference in the Sun's core, though: it is now enriched in helium and somewhat depleted in hydrogen because of nuclear fusion over billions of years. This process is gradually making the core denser and hotter, increasing the overall rate of fusion. The Sun is now 30% brighter than it was when it settled onto the main sequence.
Separation by mass of ions does occur in the surface layers of white dwarf stars; the conditions there are such that separation by diffusion can be significant. Some such stars have been seen with "metals" (elements heavier than helium) in their crusts; this has been interpreted as evidence of recent accretion of material from disrupted asteroids or planets.
https://en.wikipedia.org/wiki/Solar_core
At the edge of the core is about 70% hydrogen 28% helium with the helium fraction increasing up to 66% right at the center.
One thing I think you're missing is how overwhelmingly the cloud that formed the solar system was hydrogen and helium.
https://en.wikipedia.org/wiki/Formation_and_evolution_of_the...
The gas cloud that formed the solar system is expected to have been 98% hydrogen and helium, 2% everything else. We don't see these ratios on rocky planets like earth because the gravity here isn't strong enough to hold the hydrogen and helium, so mostly it escaped and we have a wildly non-representative sample of elements left.
Nevertheless, the standard model of the Sun supposes that, because of the very high temperature, the interior of the Sun remains mixed well enough so that the heavy elements like iron cannot segregate to form a distinct core, but only, as you have mentioned for helium, the concentration of heavy elements is somewhat higher in the center than at the surface.
However the standard model of the Sun is not very certain, because the behavior of matter at extremely high pressures and temperatures is not known well enough.
If there was an iron core, it would vaporize in the center of the sun because it's extremely hot there.
A few years ago, someone measured the energy of the neutrinos produced by the fusion in the center of the Sun [1].
[1] perhaps there are some details in https://en.wikipedia.org/wiki/Solar_neutrino
There is an exothermic fusion chain of elements that are burned in stars, the last stage of which is silicon, that ends with iron.
https://en.m.wikipedia.org/wiki/Silicon-burning_process
At iron, no further exothermic fusion is possible, so the core is no longer able to resist the pressure of collapse.
The iron atomic nuclei themselves break down, the protons combine with electrons to form neutrons where they are able, and if the total mass is below the Tolman–Oppenheimer–Volkoff, then neutron degeneracy pressure is able to stop the collapse, otherwise a black hole forms.
https://en.m.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E...
A star needs to be 8 solar masses for this to happen. Our sun ends at carbon.
They were consumed as fuel to maintain the outward pressure of the core.
Wikipedia says the TOV limit is ~3 solar masses?