When my kids were young I would make them find their own solutions to their own squabbles. If a particular toy was causing a problem I would just confiscate it until they worked out their own solution and made peace and then they could come to me and explain what solution they found (sharing, taking turns, playing together, etc.) and how they had solved the problem. That didn’t necessarily involve me asking them to apologize. More often it involved them being forced to work together to convince me that they had resolved on their own whatever problem had led to the spat in the first place. I made sure that getting dad involved was always more painful than working things out amongst themselves. I also wanted to foster a life-long sense of sisterly camaraderie because dad and mom aren’t always going to be around.
As for church? I dislike and disapprove of any sort of public confessions that put people on the spot. I don’t think the Catholic Church approach of private confessions is necessarily right either, but it is better than dragging people out in front of the whole congregation. Everyone is a sinner in some way. I don’t think it is for us to decide which sins necessarily merit public displays and which do not. That said, if some church member has publicly wronged the whole congregation in some way and feels compelled on their own to offer a public confession and ask for forgiveness I wouldn’t necessarily object. But I think that sort of thing should be rare indeed and not any sort of regular thing. I have not been a part of a church that has had any sort of public discipline and would not want to be.