You’d be better off looking back at those days AS THEY REALLY WERE. Bells were rung because there were NO electrical systems to ring a modern bell system. There were NO electrically powered loudspeakers to tell the kids “time for school” or “recess is over”. I began school in an old building in 1939 that required a hand rung bell. We moved in 1944 and the ‘newer school’ had an automatic bell ringing system. Then in an even newer Junior High building there was a loudspeaker intercom system.
Those bells were telling the time to people who in the main had NO watches. And that education system “trained” the people who built the United States of America.
In my schools we just showed up at the school yard in the morning and lined up with our classmates when it was time to go into school in the morning. Same process at lunch time. During warmer months one of the gym teachers who helped organize and "officiate" sports games for the older kids during the lunch break would blow his whistle at the appointed time to let everyone know that they had to line up to go back into school.
Call me an oddball, but I don't understand why this process would require bells or loudspeakers.
Of course, things changed in high school, but only because the students moved from one classroom to another throughout the course of the day -- and therefore the automated bell system indicated the end of each period.