However, the solid inner core does have a lower percentage of light elements than the molten outer core surrounding it. The light elements preferentially, although not entirely, stay in the molten core. As the inner core grows from the molten core freezing out, the concentration of light elements in the remaining melt gradually increases. The rising of light elements through the remaining liquid core is the main source of energy (and crucially, entropy) and driver of convection that have sustained Earth's dynamo since the inner core first formed some time in the past ~0.5-1.5 billion years (latent heat of freezing is a minor contribution). Operating a dynamo through this mechanism requires that the molten core have cooled enough to start freezing, and overall be compositionally well-mixed, without significant layering (stratification) by density, i.e. light element concentration.

The need to explain the dynamo also relates to the question of how much radiogenic (and thus heat producing) elements, particularly potassium, are actually in the core. The traditional idea, generally suppoeted by geochemistry and minerla physics, is that this amount is negligible. However, with evidence from the rock record of a dynamo for the past 3.5-4.2+ billion years, this leaves a long gap where it is more difficult to explain what drove the geodynamo.

Prior to the formation of the inner core, the compositional convection due to freezing would not have existed to power the dynamo. Therefore, a different mechanism must have powered the early geodynamo. The primordial heat left over from Earth's formation should not, by itself, be enough to sustain thermal convection for billions of years until inner core nucleation. For geophysicists, the long-lived geodynamo is much easier to explain with a thermally driven dynamo supported by the heated generated by radioactive isotopes such as potassium-40. There are, of course, other proposed explanations, such as the the precipitation of light elements near the core-mantle boundar, that is the top of the then-entirely molten core (Mittal et al., 2020; Wilson et al., 2022).

Returning to a more direct possible answer to part of u/VillageNo4 's question, the inner core might be in a 'superionic' state such that the iron metal behave like a solid, while the light elements that did get incorporated into it behave like a liquid (Wang et al., 2021 and He et al., 2022). (See also https://www.sciencenews.org/article/earth-inner-core-superionic-matter-weird-solid-liquid.) The high temperatures and pressures deep in planetary interiors can produce materials that are very exotic compared to what we see in everyday life. (c.f. Jupiter's liquid metallic hydrogen mantle and possible 'solid' core, which if it exists would not have a well-defined surface.).