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.

A horn or bell is mostly for telling other people "hey I'm here, stay out of my way and dont suddenly cross into my path"

My opinion as a cyclist is that I should basically only be using my bell on pedestrians when the pedestrians are wandering onto the bike lane. If im cycling through a shared space, I find it extremely rude to ring the bell, because it feels like I'm telling people to get out of my way, but they have just as much right to a shared path as I do. Some cyclists ring their bells because they're worried a pedestrian might suddenly turn into their path, but I think if one is concerned about that, it's a sign youre cycling too fast, and should just slow down.

With cars, I will sometimes proactively ring my bell at them if I think they're not sufficiently aware enough of me though.