I'll go down this rabbit hole for you because I was also curious.

https://www.scientificamerican.com/article/ancient-roman-lead-physics-archaeology-controversy/

All lead mined on Earth naturally contains some amount of the radioactive element uranium 235, which decays, over time, into another radioactive element, a version of lead called lead 210. When lead ore is first processed, it is purified and most of the uranium is removed. Whatever lead 210 is already present begins to break down, with half of it decaying on average every 22 years. In Roman lead almost all of the lead 210 has already decayed, whereas in lead mined today, it is just beginning to decay. (Of course, many lead 210 atoms have already decayed in this ore, too, but the supply is constantly replenished by uranium in unprocessed lead). "The longer since it was originally processed, the lower its intrinsic radioactivity," Gonzalez-Zalba says.

You won't get the same purity. Not because of worse industrial processes, but because with ancient lead any radioactive impurity normally introduced by the ore had enough time to decay.

We simply can't filter out trace amounts of radioactive material naturally existing in or around the ore as reliable as time can.

Not all of it is stable isotopes. Some of it has traces if radioactive stuff in it, since that stuff usuially decays Into lead, the highest atomic number element with stable isotopes and you might have a really decayed batch that's mostly just lead now but not lead enough for certain uses. Or it has unstable lead isotopes

Yeah all the lead atoms are the same age, whether refined or not. Does refining lead somehow make it radioactive?

Not all lead is formed at the same time, a lot of it is made by decay of other stuff and the decay chains are of different lengths, if I understand this correctly.