The .243 can be finicky if you're loading for it... and factory loads don't tend to shoot that well. I love the .243 win, but it's tedious to load for. Coaxing great accuracy from the cartridge is possible, but it's not as easy as with other rifles.
The .270 is better in this department... but still not as good as the 30-06. The 30-06 can digest pretty much any powder you have on your shelf (in one capacity or another), and 30 caliber bullets can be had from 110 grains all the way to 200's... (these will stabilize in the 1:10 twist).
The 30-06 is still the most popular rifle cartridge in the world, 106 years after its inception. It is popular because it works, and works very well... it's rarely if ever a "compromise."
I know the 30-06 sounds a bit "pedestrian" to many folks... just not as "ooh ahh" as other newer calibers... but I can't think of a better cartridge for a rifle with the "American" monicker.
And by the way... it'll kill a paper coyote deader than a stomped on frog.