Not too sure what you mean by the 30-06 not being very accurate. If you have had a bad experience with a 30-06 perhaps try a different load for that rifle. That old work horse has earned its reputation by killing more deer and elk than any of the other calibers you have mentioned. Oh and it has also taken a fare share of Japanese, Germans, and North Koreans as well.
Have you by any chance given any consideration to the 300 win mag? This happens to be my favorite caliber for big game. I will just throw out my two cents. The day may come that you are hunting more than whitetail, and this is a good all around caliber. It has a larger variety of bullet weights than the 7mm mag. Dont get me wrong, that 7mm is awesome! But the 300 does offer heavier bullets if that is what a person desires. It also has a flatter trajectory than the 308. At 500 yards you are looking at around a 54 inch drop with a 200 yard zero with the 308. So you better be spot on for your range estimation! That is also assuming you are using a 150 grain bullet. Typically for a 300 win mag zeroed at 200 yards you will have a drop of 30-35 inch drop at 500 yards. The 180 grains will usually sit at 40 inches at that range as well.
I have had good experiences with Remington 700s. I think they are reasonably priced. I have never owned a Winchester model 70 but there is no question of it being an excellent rifle too. If you do chose that caliber, I would look at the Federal fusions 150 for deer and 180 for elk. It is reasonably priced, easy to find, and performs well. Good Luck!