I have been taught (and I have read it in older Catholic books) that the bells were rung in the Mass at the epiclesis to alert the congregation that Jesus was present on the altar, and that they should stop praying the Rosary or any other devotion and adore Him, Truly Present.
Bells were necessary in the Latin Mass because many people didn’t understand the language, didn’t have missals (many couldn’t read), and needed the auditory cue so that they could be aware of Jesus’ Presence.
In the OF Mass done in the vernacular, it’s obvious to those who are paying attention when Jesus is present. Also, in the OF Mass, the people generally do not pray other prayers like the Rosary during the Mass because they are supposed to be participating in what’s going on. So no auditory cue is needed, other than the words of the priest.
However, in many parishes, the bells are still used in the OF Masses. It’s a lovely tradition, and helpful for those who lose track of what the priest is saying.
We use bells in our very modern OF parish. Whether they are rung three times, one long rrrrring, or a wimpy “tinkle” depends on which altar server is ringing them. I’ve been told by parents that some of the children are very self-conscious about being the “ringer,” and get nervous when carrying our their responsibility. So sometimes, the bells 'tinkle" instead of “rrring” or “rrring x 3”!
As I understand this tradition, there is no “requirement” for a three-times ring or any ring in the OF Mass.
I think we all need to be forbearing and loving whenever children are involved. They’re young, scared, and they will learn through practice and experience, and our encouraging words and smiles will help them to learn