First off, I would take the action out of the stock, put it in a maintainence cradle and actuate it there where you can see what's happening. The 700 safety lever has a small bb under it which detents in countersunk spots. So you should feel a definite stop in the travel of the lever. During the hangfire, if you jiggle the safety lever, does the rifle fire (or the firing pin fall)? Does it happen everytime (or very often) when you go from safe to fire??
In addition to a safety issue, you are talking about a confidence issue also. If the cure to the problem didn't jump out at me, I'd be on the phone to Remington and take the rifle to a certified smith.