I have shot an old Ruger M77 in 7mm Mag for 42 years and with one exception never had a problem with it. Those were good rifles, shot well functioned well. I finally sold it off as I have several rifles and really didn't shoot that one other than hunting.
I have several 06's, mostly Remington 1903s and 1903A3s. One is a sporterized 1903 Springfield circa 1920. It's 100% reliable and that is one I shoot for fun and hunting.
Then there are a couple Savages and a Remington 700, all heavy barrels more for target shooting but could use them for hunting. My only gripe with the Savages are the feed problems they seem to have from time to time. I would not use mine in bear country, nor would I use some of the Rem .338 Ultra Mags, same problem.
I think if I were in deer country I would most likely go with a Remington 700 classic in 6.5 Creedmoor. Great deer round, great rifle and very reliable. Or, frankly, stay with the sporterized Springfield 06!
Good luck.