Remingtons have adjustable triggers. If you set the trigger too light, you defintitely can get it to the point, where if the trigger is touched with the safety on, then the safety is pushed off, the firing pin will drop. I like a light trigger, first encountered this years ago after adjusting my 722 trigger. After adjusting the trigger and before loading the rifle, try slamming the bolt home, putting the safety on, pulling the trigger, pushing the safety off, until you are satisfied. If the trigger was never adjusted by the shooter, then unless it was set too light previous to his ownership, I don't know the answer. There are three adjusting screws on a model 700 trigger. Instructions for adjustment are available from the internet. If you have an older Remington 700/721/722 it is quite easy to grind the little lever so that the safety no longer locks the bolt, so the rifle can be loaded/unloaded with the safety on. Remington will perform this for you for a price if you go to your factory authorized repair depot, but they will also set the trigger heavy. Yes, you should use the safety, BUT DON'T DEPEND UPON IT. If you are slogging through brush, etc. it is wise to not chamber a round, but sitting in a deer stand??? I am sure as heck gonna have one up the pipe.
Bill