.300 RUM with the 200g Accubond would be a hard combo to beat for all around. Maybe a little much for deer, but for goats, sheep, elk, griz, they are tough and if your intending on builing the rifle with a brake, it will be very manageable to shoot. I weigh 135 lbs, I do not have a brake on my Rem 700 LSS RUM. The recoil does not seem to bother me, and I can still shoot sub moa out to 800 yards. If you dont feel the need for all that power, the good old .300 Win Mag would still fit the bill for everything above and out to 600 yards.
I say BS. I reject the question. No one needs settle for one gun. It's not a Ford or Chevy thing either. Going after Griz with a .243 is not like picking a Ford or Chevy. A better analogy might be 7.5 L Ford v. 7.4 L Chevy (The Chevy easily wins if you want to make lots of HP cheap). The 7mm (SAUM, RUM) is probably close to optimal for everything sans the Griz. If/when you do go after Griz, I recommend a 375 RUM.
Fads. This hyper velocity alphabet game is getting out of touch with reality. Better than half of the average guys can't shoot one of them 30 caliber boomers accurately, the other half will jar their fillings loose.
Get a plain vanilla 300Win.Mag and choose a heavy bullet for the bear. On the other hand, why limit yourself to thirty caliber? A 338Win.Mag? Might be a bit heavy for goats and sheep?
The whole question is a bit lame unless you disregard the specification for grizzly. If you have to take that into consideration, now it isn't an All Around North American Big Game rifle; it's a dangerous game rifle. A hyper velocity thirty caliber is not perfectly suitable for grizzly bear, in the real world.
I'm certainly know expert, so take this for what you will. I would check the local game laws. I'm not positive, but i'm pretty sure that in at least some areas a .338 is the minimum requirement for hunting bear. I know this doesn't answer your question, but it is something to consider if you're really considering hunting grizz or kodiaks.