We have a six slot wine rack. We have two bottles of wine and three bottles of sparkling juice (for holidays) and one bottle of water in there right now.
I'd live with it for awhile and see what develops. If you find it is a waste of space for you, then you could gut it for the shelf or cover it with a piece of artwork or fabric (to cover the emptiness). Regardless, I think living in a new space for awhile is worth doing.
I wouldn't personally use it for mail because of the aesthetics, but that's just me.
We're in a similar situation as you OP...the house we're renting has a very similar built in wine rack in the kitchen.
My solution: I use it to hold jars of dried beans, grains, rice, pastas, etc. For me, it makes things easily accessible and clears up a little shelf space in my pantry. I made labels that are on the lids, so I can see what I need and just grab it.