Why the "Mission to Earth’s core" proposal suggests using liquid iron instead of lead?
I've already commented on this before here. This will not work regardless of whether this is iron or lead or anything else. The fact that it was published in Nature does not mean it is true, always remember that.

A 108 kg ball of iron would have about 30 metres in diameter, which is not a lot, compared to Earth scales.

Here is why it will not work:


We have very massive objects on the surface. They are called buildings. They don't sink. Even though they are not that dense, buildings stand on foundations that funnel their mass to a small area, creating very large pressures. The point of the foundations is to hold the building on solid rock so they do not sink.
The author suggests to initiate the crack with explosives. Perhaps a nuclear explosion. We had many nuclear explosions in the past - they don't make cracks. They make craters.
Crack propagation is only relevant in the lithosphere. Once you get to the asthenosphere, about 200 kilometres deep, cracks don't propagate any more and the equations the author presents are meaningless. Plastic behaviour of rocks is achieved much earlier anyway.
Even if there is a crack, it will have smaller cracks radiating from it. The liquid iron, being liquid, will fill those smaller cracks and lose from the mass of the main liquid blob. With time, the blob will get smaller and smaller until it will not be able to do anything any more.
The melting point of iron is more than 1500 °C, this higher than the temperature of the ambient rocks. It will lose heat quicker than it gains by gravitational potential and solidify.
The iron will react with the rocks around it. Particularly if it is liquid, because these things are highly reactive. It will start dissolving things into it (which will counter point 5), but it will also lose iron to the surrounding rocks. It will dissolve oxygen, carbon, sulfur, possibly alkalis like sodium, maybe silicon. These are light elements that will overall lower the density of the ball of iron, bringing it to a stop.
The oxygen fugacity of the crust and upper mantle is higher than iron-wustite, which will be imposed by the equilibrium with iron. Therefore, it will leach oxygen from the surrounding rocks, essentially turning some of the liquid iron into "rock". This will also decrease the mass of the blob of iron.


You were specifically asking about lead. All points above (except 5) are equally applicable to lead.