I do shoot some loudenboomerkickenharders up tp including the .416 Rigby that does not have a recoil pad. Note twixt cost and recoil, it doesn't get shot very often.
However the two rifles that beat me up more than any other cartridges are my two Winchester M70s in .338 Win. Mag. I mention this because a friend drew and elk tag last year and had a Browning BAR with the Boss system in .338 Win. Mag. and wanted my help getting it sighted in. It had the factory recoil pad and frankly. I cringed at having to shoot that rifle. based on my experience with the M70s. I was very surprised at how well that rifle with the BOSS reduced felt recoil and shooting that .338 Win. Mag. was actually pleasant. I'm thinking of putting a muzzle brake on one of the M70s along with the Decelerator pad.
What did surprise me looking at the various recomendations was no one mentioned the 7x57 Mauser round. I have one in the M70 Featherweight in tghe earlir push feed version and with factory ammo is a real pleasure to shoot. Put a Pachmeyr Decelerator or Limbsaver (I prefer the Pachmeyr)recoil pad on one and it should be a pussycat to shoot. This is with factory level ammo. Federal makes a premium load with the 140 gr. Nosler Partition and that should put any elk on the ground if properly placed. Nice thing about the 7x57 is while at factory level it's quite mild, a careful reload can easily take it to the same level or slightly higher than the 7-08. Recoil at that point is no more than the 7-08 in the same weight rifle plus you have the advantage of using the heavier bullets in 175 gr. weight should you feel the need.
I'm currently working toward a top load for my M70 Feathersweight using the 140 gr. Barnes TSX bullet. If I get what I want, I just might take it on my cow elk hunt next January. I'll still take my .35 Whelen as back up gun rather than it's being my primary.
Learning to shoot from the left side isn't too hard. I taught myself how to do it after trying to turn to an awkward postion to try to shoot a deer right handed and spooked it due to my somewhat clumsy movements.
FRankly, if the OP really would rather go with the 06, find someone with a BAR with BOSS and try it. I have one of those shoulder pads that mount up kind of like a shoulder holster for when I shoot the really big boomers and it helps If the OP goes for the BAR/BOSS combo and the store has a gunsmith on site, have him put a Decerator or Limbsaver of the gun and he should has a package that will shoot good, be accurate and hopefully not beat his shoulder up all that much if any. My shoulder has some arthritis and the big ones get uncomfortable in a hurry. My .404 Jeffery has a muzzle brake and it does tame that one down to about 30-06 level.
Anyway that's my humble two cents worth.
Paul B.