If gravity is zero at the center of the earth, why is there a core of heavy elements, such as iron?

Alternate question for the opposite hypothesis:

If gravity is greatest at the center of the earth, as classical education tells us, why is the core not dominated by the heaviest elements (elements heavier than iron)?

I am a person reasonably familiar with technical terms, but I am not a physicist so I will appreciate answers that don't rely on equations. I am 70 years old and I want to explain it to my mother who is equally curious.

Forget about force. Force is a bit much irrelevant here. The answer to this question lies in energy, thermodynamics, pressure, temperature, chemistry, and stellar physics.

Potential energy and force go hand in hand. The gravitational force at some point inside the Earth is the rate at which gravitational potential energy changes with respect to distance. Force is the gradient of energy. Gravitational potential energy is at it's lowest at the center of the Earth.
This is where thermodynamics comes into play. The principle of minimum total potential energy is a consequence of the second law of thermodynamics. If a system is not in its minimum potential energy state and there's a pathway to that state, the system will try to follow that pathway. A planet with iron and nickel (and other dense elements) equally mixed with lighter elements is not the minimum potential energy condition. To minimize total potential energy, the iron, nickel, and other dense elements should be at the center of a planet, with lighter elements outside the core.
A pathway has to exist to that minimum potential energy state, and this is where pressure, temperature, and chemistry come into play. These are what create the conditions that allow the second law of thermodynamics to differentiate a planet. As a counterexample, uranium is rather dense, but yet uranium is depleted in the Earth's core, slightly depleted in the Earth's mantle, and strongly enhanced in the Earth's crust. Chemistry is important!

Uranium is fairly reactive chemically. It has a strong affinity to combine with other elements. Uranium is a lithophile ("rock-loving") element per the Goldschmidt classification of elements. In fact, uranium is an "incompatible element", which explains the relative abundance of uranium in the Earth's crust.

Nickel, cobalt, manganese, and molybdenum, along with the most extremely rare and precious metals such as gold, iridium, osmium, palladium, platinum, rhenium, rhodium and ruthenium, are rather inert chemically, but they do dissolve readily in molten iron. These (along with iron itself) are the siderophile (iron-loving) elements. In fact, iron is not near as siderophilic as the precious metals. It rusts (making iron is a bit lithophilic) and it readily combines with sulfur (making iron a bit chalcophilic).

This is where pressure and temperature come into play. Pressure and temperature are extremely high inside the Earth. High pressure and high temperature force iron to relinquish its bonds with other compounds. So now we have pure iron and nickel, plus trace amounts of precious metals, and thermodynamics wants very much to have those dense elements settle towards the center. The conditions are now right for that to happen, and that's exactly what happened shortly after the Earth formed.

Finally, there's stellar physics. The Earth would have a tiny little core of rare but dense elements if iron and nickel were as rare as gold and platinum. That's not the case. Iron and nickel are surprisingly abundant elements in the universe. There's a general tendency for heavier elements to be less abundant. Iron (and to a lesser extent, nickel) are two exceptions to this rule; see the graph below. Iron and nickel are where the alpha process in stellar physics stops. Everything heavier than iron requires exotic processes such as the s-process or those that occur in a supernova to create them. Moreover, supernova, particularly type Ia supernovae, are prolific producers of iron. Despite their relatively heavy masses, iron and nickel are quite abundant elements in our aging universe.