I've noticed a change in the games I refer over the years, I assumed it was just that my tastes have changed, but I really think it is the games themselves. I used to really favor the games with multiple branching paths, where each branch really changed the story, but these days I mostly play games with very limited branching. BUT I went back and played Come See me Tonight 1 and 2 recently, and I still loved them. I think it has more to do with how modern VNs are written by 1 or two people, so they just can't put as much time and effort into deep branching narrative, whereas those games i so fondly remember were handled by studios with many more employees, sometimes as many as a dozen writers, so they could focus on deeper, more engaging stories for each branch.
Single playthrough, and if there are branching paths I bust out the Notepad++ and merge them. Not getting me to play through your crap twice, lol. The only time I'm ok with branching is if there are completely different storylines that don't overlap.
Comes back to the quality of the game. If it's really good, the branches and paths are likely to be different and it's like playing a different game, or seeing the same events from different perspectives. If I'm really enjoying a game then I'm happy to play it on different paths, sometimes straight away. Or I can come back to it after a few months and try a different route. If it's poor then once is enough and see if there's anything in the gallery which catches my eye.