TL, what happens to people who don't drink at all for whatever reason? Do they still have to turn up to these drinking parties once in a while?
Generally everyone has at least one drink so they can participate in the initial kanpai (cheers!). You can ask the sever for a mixed drink (like a haichu) and quietly ask them to make it light. After that you're generally off the hook as long as you're drinking something (even non-alcoholic). And usually the manager is happy as long as there are people drinkwith him, and there always were. People who aren't going to drink much sit away from the managers!
I have heard about nasty situations like Mangamoz is talking about but never run into them in Japan, maybe because I was there!
If you have medical reasons, that's understood, or if you're the driver (but it's almost always by train then short walk). Honestly, I never ran into a situation where someone there just refused without a good reason. I specifically asked about people who don't go, which is how I know, but didn't ask about people who go but just refuse the first drink without one of these reasons.