I'm sure the .300 is more than capable of anything in north america.
So is a 30-06 or 308, you just have to get a little closer
I'd advise working up to the 300. A 308 will do anything at 400 yards that a 30-06 will do at 500 yards and a 300 will do the same at 600 yards. Are you good enough to take advantage of the 300's capabilities. A 308 is capable of doing more than most shooters can take advantage of.
I'm not opposed to someone owning a 300 mag, have one myself, but it is not a good choice for a new shooter for many reasons. You'll shoot a 308 a lot more because you'll find it more enjoyable and more affordable. After you master a smaller cartrdge you may find you don't really need a bigger one.