A .223 wont kill a bear, hahah!!
Tell that to inuit hunters, .223 is a cheap, common round and they hunt with them. Dont forget that inuits hunt polar bears, the biggest, strongest, and only carnivorous bear out there.
While it may not kill as fast as a .375, or a few others, dont believe the hype about needing a 105 mm howitzer to bring a bear down. Ive witnessed black bears, and deer being dropped from 300 meters with a .223.
Sure its good to carry something that inflicts a strong initial impact, but its also important what the bullet does after that impact, it displaces fluid, the small high velocity .223 and 5.56 rounds tumble, and break up while traveling through flesh. The result is a huge dose of displaced fluid, torn up muscle, and internal organs.
If your hunting a bear and want to save as much meat or organs as possible then you want a big heavy stable bullet, the bigger bullet will travel straight through and make a nice neat big puncture.
Unless you shoot a bear in the leg, or miss the midsection entirely with a .223 your going to inflict a fatal wound on a bear.
Remember they are living breathing mammals not armored mechanical tanks. If you shoot a bear in the right place with a .22 you can drop them.
Truthfully a shotgun will penetrate alot less, and do less damage by far internally than a .223.