In all probability the factory bedding won't need any work. The forward recoil lug pretty much insures that. Don't know if your rifle has it, but Winchester was even bedding the lug on the under side of the barrel on Alaskans. A really good idea.
You may however, want to consider something I've done to all my Walnut Stocked guns, is seal the hell out of all the inleted (sp?) areas to prevent not only moisture but oil saturation into the wood. I've used three coats of a quality Polyurethane (not water based) and have rarely needed any touch up for years after. Getting one wet in a downpour is just a way of life and I've never had one change point of aim/impact from a stock warping.
I'm a Model 70 addict. The older the better. Enjoy your M-70, enjoy the .375, they make a great combination.
Appreciate the info! I usually try and seal up wood stocks, both the outside and inside with Birchwood gun wax (just did a CZ 455fs in 22mag last night). I like your idea of the polyurethane on the inside of the stock.