I haven't hunted with a bolt gun, so maybe I'm out of my element here, but when I'm hunting with a shotgun or doing move and shoot competitions with my AR, the safety goes on until I'm getting on target. If I fall and faceplant, I don't want the firing pin dropping even on an empty chamber. I specifically only buy guns that I can easily take the safety off while shouldering so at this point it's just part of my muscle memory. That said, I haven't done hunting that's hours of walking around with the gun slung so I might not keep it chambered in those cases, but if I'm going to be thumbing the safety either way why not have the safety on? And hey, having the rifle already cocked with an empty chamber saves .010 seconds on cycling the bolt so it's a benefit /s.
I do frequently have my Tikkas off safe at the range, but that's because when the range goes cold I'm dropping the magazine and opening the bolt and I'm not even sure if I can actuate the safety with the bolt open. IDK maybe I should turn it on even then if I can, but with the gun on the bench with the bolt back, chamber empty, and magazine out it has pretty thoroughly been shown clear.