A good example and an ordinary event you've chosen which many people can identify with. I wonder a little further regarding the relationship between you and the other woman.
It shimmers through that you probably would have liked to be of more support during her unexpected announcement that she cannot cope with her daughter - do I perceive that correct?
I know this feeling myself when someone blurts out something which he or she feels helpless to. My first reaction is similar to yours. I get silent. Don't want to say something trivial so I rather shut up and let silence speak for itself. Sometimes, this gets the other to open up and to tell more. Only nodding and listening activates what else will come up. I felt a little disappointed while reading that this moment of the two of you was interrupted and didn't get a chance to develop.
When I am in the shoes of the mother who feels frustrated and helpless, I like it when someone who I share this with shows me patience and assurance that he is not uncomfortable with me in this moment. I like it when this interlocutor starts to ask me questions. ... Requires a real interest in me finding my potential of solving a problem of mine.
... Also, what had helped me when someone was confirming what's difficult for me. My brother back then told me that I way too much discuss with my three year old. That hurt in that moment but I knew instantly he was right and that I did not allow myself to be the authority in this relationship.
I took on consulting and talked with my colleague about this matter. What I found out through my own counseling other people is the surprise when I spoke out what others only may think: "Is it possible, that you don't dare to take on full authority?"
Did you get the chance to mirror her daughters well response to you and that she might behave towards others in a better way? That - I experienced - gives so much relief to parents as they might think that their child behaves towards all people in the same way. ...
Indeed, I felt I run out of time, and when we found ourselves alone once again I honestly thought she was going to elaborate more about her feelings. But she didn't. So, I just dropped it. Despite of everything I had in mind to discuss with her. As I felt that, due to her vulnerable state, she might have taken it as criticism.
As excited as I was when I saw how Lilly reacted to my request I soon fell into despair as I watched her bed time behavior. I do wonder how Lilly is going to be when she becomes a teen. Her last one was having mum and dad turning the world cup game off to watch her dance. They diligently did it. And I really don't know why!
I suppose this is because of the uncertainty and the projection of what adults believe themselves - and thus their daughter - to be capable of. ... It could also be that they did that because they were witnessed and did want to spare the stress saying "no" to her...
... As I read it so far, a vicious circle seems to have developed, so that parents and the child can come into less trusting contact at all. The child feels, the parents are tense.
It is not easy to criticize... Therefore the question would be interesting whether your criticism could not be one that the woman already heard elsewhere and is now confirmed by you. There is a risk of making oneself unpopular and even provoking unfriendliness and counter-talk. To accept such feedback in a positive way at the moment of its creation is rather rare. I have seen that such things usually happen afterwards and are not necessarily brought close to the person who gave the feedback.
Didn't you feel the same way when you got criticism, you couldn't respond positively to it at that moment, yes, you even got angry and yet days or weeks later you found something helpful in it?
I once had a situation with a friend in which I asked her what motivated her to formulate the demand towards her son (because I felt a lack of trust in her son). Thereupon my friend reacted very unfriendly, as my question had offended her. But if my objection led her to think about my question later on, that was the price for bringing her displeasure upon me.
I must admit, however, that my question was coloured by the fact that I did not like what I saw and that my disfavour was the cause of her reaction towards me. If I had been less clumsy and had cultivated openness to results instead of my judgement, the situation might have been less uncomfortable.
There's no guarantee, but there is a probability. Sometimes a critique goes well, sometimes not...