I can't quite see the backing board to tell its age - is this a modern piece or an antique? If it's an antique, put the doors back on it and sell it. If it's a modern piece, it's ok to paint.
I like oak bookcases and rooms that aren't too matchy-matchy so I would not paint it. I think it looks good as is. I would line the vertical books by size to give in a more even look (smaller books together, taller books together etc). The horizontal books (Dr. Seuss, maybe) look so much more organized because they are all the same size.
Oak can look great with the right material. I would not think the oak would be a problem at all. A textured solid or a simple print would look good. Adding material to oak would be a very country look.
With the assortment of kids books, it could also look busy if you added a patterned background when kids books have so much pattern to begin with.
I like oak bookcases and rooms that aren't too matchy-matchy so I would not paint it. I think it looks good as is. I would line the vertical books by size to give in a more even look (smaller books together, taller books together etc). The horizontal books (Dr. Seuss, maybe) look so much more organized because they are all the same size.
Going to reorganize books now and see how it looks. I agree with not too matchy. There is no way the oak would look good with a material background is there?
I'd try before painting it. depending on fabric I think it could work unpainted. I like the oak personally
This was going to be my suggestion, except stencil poster board instead of using fabric. I couldn't find fabric I wanted, so stenciling was my solution. But either one make a huge difference.