I'm not going to kid myself that I can make the wood as smooth as metal. That's a huge amount of work and arguably more effort than scratchbuilding in a more suitable material. I'd like to get rid of most of the grain though, and that means a coat of sanding sealer.
For those not familiar with this magic product, it's basically a big can of cellulose dope with talcum powder mixed in. Paint it on the wood, it soaks in, raises the grain a little, fills gaps and can be sanded smooth within a few minutes. Apply enough coats and do enough sanding, the results are smooth as glass.
It also smells lovely.
Anyway, I treated the outside of the frames, but not the inside. Once smoothed, I sprayed both with matt black paint. For once I remembered not to overdo it with the hairdryer as this can activate the cellulose which bubbles up in the paint. Much swearing then ensues and I'm never patient enough to leave it alone and come back later.
It's not obvious in the photo, but the untreated sides aren't nearly as smooth as the sanding sealed one. They will do the job and after a light sand aren't terrible at all. I will be sealing all the other bit of wood though.