The Sierra Game King HPBT is a 165 gr bullet with a unique crimped meplat that delays expansion for deeper penetration for lack of a better way to describe it. In every conversation I've ever had with Sierra, bullets constructed this way are described by Sierra bulletsmiths as unnecessarily tough for whitetails, especially the 165 gr bullet at 500 yard .308 velocities. The 165 gr Game Changer that was suggested by someone previously is the toughest Sierra bullet design, as it has a thicker jacket and harder lead alloy than any of the other Sierra bullets. According to the bulletsmiths at Sierra, Gamechangers are designed for high BC and deep penetration on big game, overkill for most whitetails at low velocities as well.
Not exactly the same, but similar performance can be expected comparing .308 vs 7mm08. From my 7mm08 at ~2650 fps muzzle velocity, the crimped meplat 160 gr SGK HPBT at <10 yards put a baseball sized exit wound behind the shoulder of a 120# doe. I would not hesitate to hunt bear with this bullet, but I probably wouldn't do 500 yards. B.C. leaves a lot to be desired for long range.
From my 7mm08 at ~2600 fps muzzle velocity, 168 gr Berger Classic Hunters have given me golfball to baseball sized behind the shoulder exit wounds on six 150-200# whitetails at 50 to 110 yards. Massive blood trails were obtained on the couple that bounced ~25-35 yards. I wouldn't pick this bullet for bear, but 500 yard whitetail shots at those low velocities should work pretty good.
No experience with ELDs on game at all.