Well, first, it is not a .308 Win Mag. It is a .308 Winchester.
Yes, the .308 is more than adequate to drop any animal you would find outside of Africa and larger whales, IF you put the bullet in the right place. Ammo is reasonable, caliber is more accurate than you are.
In the right rifle, can reach out effectively to 900 meters are so. Which MAY be too much for a home defense scenario- I do not know where you live, but this can shoot thru a couple of frame HOUSES.
SIG makes good guns, the AR style will allow you add whatever sights. lights, bells and whistles you want. Pretty new, so not a lot out there to choose from, and when you add sights, will pop that $2k limit.
But what the hell, you would probably just waste the money on other frivolous things anyway- like food, shelter, clothing....
If you like it, can afford it, and it fills your desires, go for it.
PS- you better check the legalities of taking a rifle into Yellowstone, or one of us will be buying that rifle when it gets auctioned off by the National Park Service. And unless you are running an Extra Full choke (used by very few turkey hunters) your shotgun is likely to handle slugs just fine.
And not that you asked, but my defensive firearm of choice at 50 meters or less IS a shotgun. About $200 instead of $2000.