Grey usually means the protective coating has eroded in the weather and you are looking at bare wood. Black marks are a fungal stain that can occur under oiled wood used outdoors, especially in warm humid conditions. I have also seen this black stain on varnished hardwood on boats.
If it was me, I would sand it back and re-oil, or rewax if you prefer.
I have been using Linseed oil on outdoor hardwood, and it seems fine. It is quite slow drying, though, and has a pleasant smell on warm days. I believe Tung Oil may be quicker to dry. Danish teak oil is mixed to dry fast but I do not think gives good outdoor protection. I have not had the fungal stain, but I used Cuprinol Clear preservative on the timber first, and mine is a dark mahogany which to my eye would look OK even if stained.
You can try bleaching the stains after you have sanded it, that might work. You can also use Ronson wood dyes on the bare timber after sanding, if you use a dark one then any unevenness will not show.
My softwood items I use a water-based shed and fence stain on, which is very quick and easy to apply, and leaves an opaque waxy, water-repellent film, and I have also used it on teak trim. I use dark colours.
Honestly, I think any timber product used outdoors is going to age with exposure to the weather, it is not going to look like a french-polished sideboard.