> Some cyclists ring their bells because they're worried a pedestrian might suddenly turn into their path,
This is wrong - on mixed use paths, it is customary and proper to announce "on your left" when passing, and a bell is a nice alternative. Even cycling slowly pedestrians can do some very erratic things, and moreover are very surprised when cyclists suddenly appear on their left (and may do something dumb in surprise!).
> This is wrong - on mixed use paths, it is customary and proper to announce "on your left" when passing
This is neither customarily nor regulatory uniform. There are mixed-use trails near me where bells are required. There are some trails where most people use a bell, some trails where nobody uses a bell, and some where there is a mix.
In my personal experience, the ratio of bikes to pedestrians and the purpose of the trail greatly affects how people tend to handle this.
On the bike trail it is crucial to do a shoulder check when changing lanes. Some people get "in the zone" and ignore all other traffic in the singular pursuit of the shortest times. They will get very very angry if you get in front of them, if they spot you at all instead of just slamming into your rear tire at full speed.
I personally can't stand to ride without a mirror for situational awareness (or, if on road, a mirror and also radar).