Do horns and bells really prevent accidents?

In order for e.g. a horn to work you need enough time that the driver processes the situation and decides the horn will communicate something AND enough time for the pedestrian or whatever to process that and react to it. Generally it's a lot easier just to press the brake, and more importantly be travelling at a speed and in a manner where the brake is sufficient.

Structurally, we'd be much better off reducing conflicts between the different tiers of users. I.e. properly segregated infrastructure for each class of vehicle.

Bicycle bells are mostly for warning pedestrians when approaching from behind and passing on shared-use trails. I ride on shared infrastructure and cannot afford to build new infrastructure when my town will not. Not warning a pedestrian when approaching from behind introduces the possibility of collision if the pedestrian makes a sudden change in his walking course. I typically use this etiquette:

Passing a single pedestrian or runner on a quiet day: no bell, coasting for a short bit with a loud free hub (the rotating ratchet element on the rear wheel) alerts the pedestrian to my presence.

Passing a runner: normal ring from a distance so they have knowledge that the bicycle is passing

Passing a cyclist: one loud ring from a distance

Passing a pedestrian walking a dog: two loud rings, one far, one close, so that the pedestrian is aware of the approaching bicycle and he can prevent his dog from running at me/colliding. Many dogs do seem to enjoy a bicycle chase.

Antisocial pedestrians (i.e., walking side-by-side such as to be blocking the path in both directions, preventing the bicyclist from passing): several loud rings of the bell until the antisocial activity has abated. Announcements in my local tongue (not English) that they impede the flow of traffic.