No, there is no reason why you can't apply linseed oil.
But here is why I wouldn't. Your stock looks fantastic. The 70 year old original finish is holding up, and holding value. Once new oils are introduced and dried, the risks of grain definition loss and overly glossy appearance arise. And once a gun looks freshly 'refinished', it looses appeal and value to most C&R collectors.
Does that mean that I don't own refinished guns? No. Some rare specimens must be accepted as refinished or not, if I wish to add them to the collection. But I simply pass on the more common refinished items, at almost any price, because original and more desirable examples can still be found.
Below is one of my Remington 03-A3's. Wood grain is still open, cartouches still look fresh. It is not excessively glossy. It will be preserved exactly as it is supposed to look.
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