I'm in the same boat as OP, looking for a new 06. Probably over thinking it but the EW has a 22" barrel instead of a 24". Which has its own pros and cons. But if one liked the 22" barrel then why not go with a 308 because it preforms better out of a shorter tube plus you get the shorter action and weight savings that goes along with that. Not to mention less recoil. But I really like that you can go higher in bullet weight with the 06 over the 308. If I was to only hunt whitetails then I would probably stick with my .270. But I don't, and I find myself in the 30 cal camp for Elk but I am not a fan of the mags. I know a 308 is plenty for Elk at reasonable distances but why not take the extra speed and thump of the 06 (even in a 22" tube). I seem to land on the 06 because I like the bigger case over the 308 even in a shorter but easier to maneuver 22" tube. Not to mention a shorter overall length which would make the gun better in timber but can still reach out in the open. At least this is what I have been mulling over in my head. Curious to hear what others think about this.
On a side I read a Chuck Hawks article a long time ago about having a 270 and 06 as a great all purpose two gun duo (not the end all be all or the only). But where both calibers excel at what they do, but one can back up the other just fine.