You know what? I don't mind if it's the placebo effect fixing my headache, or the Excedrin (or whatever). I feel similarly about sleeping pills. I know, especially with sleeping pills, that it's probably much more the fact that I've "primed" my body to expect sleep because I've done something that I know & believe is going to make me sleep, than the pills necessarily actually working. But you know what? At the end of the day I want to fall asleep, so whichever it is that helps me do that, I'm okay with that. It might all be bunk but the bunk seems to be working for me anyway.
That being said I generally don't get hangovers. I don't drink to excess a lot, and when I do, I almost always consume B vitamins (and sometimes some food) afterwards. The food prevents a sour stomach and the B vitamins seem to take care of everything else. Last time I was hungover I'd been drinking (and in between tipsy and drunk) for 12 hours. Not to mention walking and sweating in the summer sun for 2 hours. I don't think anything would have stopped that one. I threw up in an empty moving box.
I sometimes get migraines. I've had migraines that lasted 3 days, and some that didn't. I'd rather take the medicine and hope it helps than wait for three days. But like, if your pain doesn't last very long in general without medicine, than I can see why you wouldn't bother. It's not like NSAIDs are good for your stomach in any way.