One thing I don't understand. Surely the lead atoms are much older than 2000 years - it is not like the Romans "created" the lead via nuclear fusion. The lead atoms must date back to the formation of the earth and beyond.
Therefore, the Romans merely concentrated these lead atoms into one place. But why hadn't all the unstable lead atoms decayed already long before? Why does it matter if the lead ingot is ancient or not, the actual lead matter is very ancient regardless?
Pb-210 is a lead isotope that's a byproduct of uranium decay. When mined, all lead contains some Pb-210 as a result of U238 decay[0]. The lead can be refined to remove the U238, but not to remove Pb-210. So all 'fresh' lead contains some Pb-210 with a half-life of 22 years. If you use this for shielding in your experiment, you have to deal with its natural radioactivity.
(Now, what confuses me is how the Romans removed U238 from lead, because they must have done so in order to end up with lead that could decay over 2000 years. Maybe it happened as part of the smelting process.)
Most old lead objects are cast, casting is a process where a liquid is poured into a mold where it is left to solidify. Presumably because both the melting crucible and the cast are under the influence of gravity while the metal is in the liquid phase the heavier elements will sort themselves to one end of the end-product, maybe even into the flues that will get cut off from the final product.
It would be much harder to keep the U238 evenly distributed throughout the lead during such a process.
Natural lead occurs along with small amounts of uranium and thorium. These radioactive atoms have radioactive lead isotopes in their decay products. So natural lead is constantly replenishing its levels of radioactive isotopes. When lead is mined, it is separated from the uranium and thorium, so no new radioactive isotopes are produced. You just have to wait for the radioactive lead present to decay. This can take 2000 years evidently.
Lead that has just been dug out of the earth was exposed to radiation from other minerals and cosmic radiation until very recently. On the other hand this batch of ancient lead was at the bottom of a lake for 2000 years, so it had time to lose some of its radioactivity while being shielded by water.
Therefore, the Romans merely concentrated these lead atoms into one place. But why hadn't all the unstable lead atoms decayed already long before? Why does it matter if the lead ingot is ancient or not, the actual lead matter is very ancient regardless?