I've been using thinned Titebond II for years, usually about 50/50, sometimes down to at least 25/75. Mostly use on my canes, and quite pleased with the result. I would think you could paint over it, with some type of preparation, but have never tried, so can't really say. But I think if I wanted to paint over it, I'd try thinned Titebond II first, with color added, rather than regular paint.
My canes are all meant to be used, so the finish is plenty smooth enough as far as I'm concerned. And so far, has held up quite well.
And, if I want to glue paper down to wood, I find the thinned glue does well for that, soaks the paper, and when it dries, the paper shows the wood grain. You do have to be a bit cautious tho, when the paper gets soaked, if it is not quite where you want it, trying to shift it can tear it every easily.
"It ain't what you're told, it's what you know." - Granny Weatherwax
Gather the villagers, pitchforks, torches; we march at dusk!
Some days, the supply of available curse words is insufficient to meet my demands.
.....Call me a craftsman, artisan, or artistic, and I will accept that. Call me an artist and you will likely get a quite rude comment in return. I am not a @#$%ing artist.