Lead is essentially useless as a shield for neutrons that are below the minimum excitation energy of a lead nucleus. Elastic neutron collisions with lead leave the neutron energy essentially unchanged.
I assume this is why an alloy of lead is used in practice. Still doesn't change the fact that graphite is a moderator not a shielding material. Also, structural materials in reactors are usually invisible to neutrons and a sandwich of materials is often used. Different layers do different things. Usually, one layer of shielding and one layer of a material that isn't impacted (much) by neutron flux for structural strength.
There is a rabbithole for almost all of these material choices, especially in nuclear. Not going down that rabbithole in a discussion targeted at folks who don't spend their lives working in nuclear doesn't make that person wrong. It makes them an effective communicator.
PS Lead is a very very common shielding material in nuclear.
A moderator is a neutron shielding material, since it removes energy from the neutrons. That's what moderation is all about. Water is a much better moderator, but graphite still performs the function.
