I own a .270. If you were going to own one rifle and hunt "everything in North America" with it the .270 is proven to be able to do just that.
However it starts to suffer from a reverse Goldilock's syndrome. While it is capable of being used as a varmint rifle it is clearly overkill for it. The same can likely be said for antelope and whitetail under the vast majority of circumstances. There are lots of sub .270 rifles that will be adequate for that type of game (for me its the .243).
Let's get to the sweet spot: Elk. I think the .270 might be the perfect elk rifle.
Then you get into the "its adequate but marginal" category. Moose, aggressive bear, etc. Yes the .270 is plenty but if I was in an area where aggressive bear were likely to be encountered I would like something bigger. For the record, and for me, this means .375 Holland and Holland. Plenty of people will point to the .30 calibers, a .338, or something else here but that's all up for debate.
When you are discussing augmenting a .308 with a .270 I am kind of left wondering about the why. The .308 and .270 seem to have a tremendous amount of overlap to each other and I am not sure what one is going to do that the other is not going to do.
But, in the end, this was not the question the OP asked. The OP stated he wanted a no-frills .270 in the sub $700 range. To me the no frills price range rifles are defined very well today by Ruger, TC, Savage, and Mossberg in their various lines. And for the record many of these "no-frills" rifles boast reported accuracy out of the box that should shame their "lots of frills" high end brothers.