For me, it’s more about bullet quality than looking at construction alone.
So for my 7x57mm Mauser, I use Winchester Super X cup & core bullets (now discontinued) and Nosler Partition bullets. Both of these are lead cored.
For my 12 gauge, I use Rottweil lead birdshot and copper plated Federal OOO buckshot.
In a .30-06, I’ll use Remington Core Lokts and Sako Hammerheads. Both of these are lead cored.
In a .375 Holland & Holland Magnum, I’ll use Trophy Bonded Bear Claw bullets and Rhino Solid Shank bullets for hunting most game. Both of these are (at least partially) lead cored. When using the same caliber on elephant, I’ll use Norma 300Gr monometal solids. I believe this is all copper or brass.
In a .600 Nitro Express, I’ll use Labor Fur Ballistik copper jacketed TMR (soft point) and VMR (FMJ) bullets. Both of these are lead cored.