Will respect, this is a very partial account. Not all of the things I mention will be relevant, but there is not enough information in your post to answer fairly, so I'm jut throwing ideas out there.
Where was your daughter going? If the children were staying because she had an important appointment or a big social event it is different from if she wanted a night off, although even then, she will have made plans around not having the children, and you let her down at the last minute. What would you expect an apology for? I would apologise profusely if I'd agreed to look after children then rescinded the offer because I couldn't cope, and ruined someone's evening, but I wouldn't expect them to apologise to me.
Did you want to have the children? Was there no way you could distract the little boy? Children often cry when their parents leave, but settle down quickly when distracted, particularly when they are in the care of someone whose company they enjoy and where they feel safe.
Does your daughter 'put your grandson into activity days' because he enjoys them, because he is a handful, or are you suggesting (as your choice of words maybe suggests) that she doesn't want to spend time with him? If he is 'difficult', maybe your daughter needs support? What about her husband? Does he work?
Does your daughter know that you use physical punishment on her children? My mother might have described her approach as 'a tap on the bottom', but that is not what my sister and I remember about it, and was very clear that under no circumstances was she allowed to hit my children on the rare occasions she looked after them. I am reluctant to believe those who claim to 'tap' children, as there is absolutely no point in doing that. If you believe in physical punishment the child has to dislike it, and therefore it has to hurt to be effective (if it is ever effective, which I doubt, but that is a different debate). If your daughter doesn't approve of her children being hit and you do it anyway, it suggests that you see your way of doing things as the only way and are prepared to over-rule the parents, so maybe there is a disconnect between your own and your daughter's views on behaviour (yours and the grandson's).
What made you think the boy had been missing his mum? Many, if not most children are at school five days a week at five (most start at four) and yes, they get tired, but it's quite unusual to assume that bad behaviour is down to missing their mum. Does he see his father? Or miss him? You do come across as if you blame your daughter for her son's behaviour - do you disapprove of her working? Maybe you think her husband should cut his hours so they can better share the care of their children? Either way, it is up to them how they divide the childcare responsibilities, and you are probably best advised to keep out of it.
To be honest (and as I say, only given the limited information in your post) it sounds as though you are right that the situation was out of control, and that you weren't able to take control of it. As the responsible adult, it is vital that you are in charge, so maybe it's better if you don't look after the children overnight in future. It can be unsettling for children to feel that there is no responsible adult in the room. Of course it should be up to you to fix problems as they arise - who else is going to do it?
Sorry if this is not what you wanted to hear. I realise that it sounds critical, but there seems to be a subtext of resentment in your post, and your grandchildren will pick up on any reluctance to have them and to take responsibility for being in charge.
