I've been a cyclist in SF and in Amsterdam, both for many years.

In SF I used my bell much more aggressively. It was mainly for cars, if I'm in or entering their blind spot and my spidey sense tells me they are considering an action that places me in danger. For example, we all know when driving when the car in front of us is thinking about merging, even before they indicate (often I feel like I know before they do). I also used it for pedestrians stepping out into the street who are maybe looking past me for oncoming cars but somehow don't see me, or when approaching 'blind' situations like a sharp corner, a driver pulling out of a driveway but there is a tree between us, delivery drivers stepping out from their truck, etc. I can't say how many accidents have been prevented (the person may have eventually looked and seen me), but I can say that my bell has triggered people to look and see me earlier than they were going to had I not rang it.

In Amsterdam my bell is used much more sparingly. It's mostly for tourists stepping into (or considering stepping into) the bike lane. If they are already in the bike lane, I almost always prefer just to slow down a bit and dodge them, as ringing the bell often triggers a deer-caught-in-headlight moment or erratic behavior, which increases the chance of an accident or that I have to come to a full stop. The other situation is to express dissatisfaction at cars blocking bike lanes, cars/bikes not yielding, drivers blocking intersections, or other dangerous behavior. This isn't preventing an accident but I'd argue it is still important, as social control affects how often we make bad decisions. Outside the city I also use my bell to let other cyclists know I'm passing.

So yeah, I'd say bells prevent accidents, but obviously not as well as good biking infrastructure, where pedestrians, bikes, and cars have clear separate spaces, and visibility of cyclists to drivers is high.