[Chemist1961]

Depending on the grain in your oak it may be impossible to make it dissapear under paint. While it seems like a waste we have painted over 30 year old oak cabinets and reduced the natural look for a bright clean kitchen.
Seems kind of odd to lose the oak elsewhere and install more wood in the bathroom, but I have a hunk of butcher block that I want to use for an island breakfast bar. I think I will stain it dark and seal it with varathane or similar.....

 

[Chemist1961]

Nice oak but a lot of it. I just put down cherry flooring, then the wife picked a dark brown paint to do a pair of accent walls. The brown pulled up on the flooring and changed the whole room. My cherry is a similar shade to your oak.
Your's is pretty stark, very clean but stark but even another shade of wood will look stark against all the white. Why not paint the walls to tone down the contrast before you tackle all the wood