I don't think things have changed as much as you seem to think they do. When my daughter is old enough that she might be under someone else's supervision or otherwise playing with the neighborhood kids outside of my supervision, if she does something she shouldn't, I hope whoever is supervising will discipline her. However, they better not lay out any physical punishment. This is the same thing I've heard my mother say. She expected others to discipline my brother and myself when we weren't under her supervision but, if anyone laid a hand on us, she would have come down full force on them. In my opinion, that's just appropriate. It was when I was a kid and it still is when I am a parent. You discipline other kids but leave physical punishment decisions to the parents.
As for kids crying in public, I agree. If you can't get the kid to calm down quickly, then you get the kid out of the public setting as quickly as possible. Of course, if you can't carry the child and they are doing one of those laying on the floor tantrums, your options may be limited. I remember seeing tantrums years ago, though they were not nearly as frequent as I see today. Maybe I didn't notice them, maybe I'm now in places with more people so I should expect more, maybe they really are happening more often. I can't say. I do know that I am never comfortable when Shayla starts crying in public and I will always try to either get her to stop or get her removed from the public place as quickly as possible when she does. You might hear her cry if she's at a store or restaurant with me but you won't hear her crying for long. She will either be gone or calmed down. I think more parents are like this than you realize. It's the kid throwing a tantrum for 15 minutes that you notice, not the many who don't throw any tantrum or who cry for a minute or less before calming down or being removed from the situation.