I have had some very accurate 30/06 hunting rifles. In fact, one of the most accurate hunting rifles I owned was a 30/06. However, in any accuracy game, I am convinced the 308 is the more accurate of the two. I HAVE shot both cartridges from rifles built one the same action and, in a couple of instances, using the same barrel. The 308 was always better.
Don't get me wrong; the 30/06 didn't suck. It just didn't shoot quite as well as the 308. I have had 308's which were legitimate 3/8 moa rifles or even a bit better. I have not managed 1/2 minute with a 30/06 on a consistent basis. I have gotten to around the 5/8 level but that's about it.
In the hunting field, all of this is, of course, meaningless. A barely sub-moa rifle is more than adequate at any reasonable range.
One thing that being connected to the internet has done for me is to destroy my confidence in my shooting ability. I'm just not that good! At three hundred meters, I often shoot groups as large as five inches with various hunting rifles. In fact, with my best target rifles, I have had a shot land four inches east or west of where I expected it to go. The average internet shooter NEVER does this so I've spent the last ten years in a state of depression over my apparent inferiority.
On the other hand, the list of rifles with which I have shot sub 1/2 moa groups is long and varied. The list includes such unlikely candidates as a Marlin 45/70 and a #4 Lee Enfield so maybe there is some hope for me. Regards, Bill.