It certainly possible to allow oak to go silvery grey, but:
-On some aspects that receive no sunshine it could go black or green
- It depends on the joinery design, traditional joinery that had no provision for draught sealing or modern locks could cope with a high degree of movement without any problems. If your French doors operate with a multipoint lock then you may find there is a limit to how much it will cope with such movement.
-oak left untreated will initiallly suffer from water staining on the faces, ingress of water into the joints resulting in the joints going black due to tannins, end grain will soak up a lot of water so you may get excessive swelling at the bottom of door stiles etc.
You could try treating the doors with a water repellant like Thomsons which will stop the water staining in the short term and let the sun do its work killing off lignin and turning the surface grey. I would recommend using end grain sealer on bottoms of stiles etc. There is no clear treatment that will protect the oak and allow it to go grey
I now offer customers a low build light oak finish which was actually designed as a decking and cladding stain. It has a matt finish and does not look like the high build transluscent finishes that joinery companies generally offer, which I think look too plastic.
In the joinery industry, the colour 'light oak', is generally accepted as the lightest tint that will provide an acceptable UV protection. Whilst there are clear finishes that have UV inhibitors, they are unlikely to last more than a year. Danish oil is not really worth using outside, it will protect for a few months but will get burnt off quite quickly.
