It's true that cores are swaged, but so are whole bullets. Take a look at the gray bullets on this Hornady list
, or at the Corbin bullet swaging site, here
They aren't all pure lead. Hornady or Speer used to claim theres had a couple of percent Antimony. Corbin sells pure lead wire, but he also has a calculator for figuring pressure for swaging different alloys up to about BHN 8-10.