Giving a child some space to decompress after school is NOT pandering to them, @Pricelessadvice - it is seeing what an individual child needs, in a particular situation, and giving them that.
As an adult, we can do this for ourselves - I am an introvert, so if we’ve had guests round, I know I need a bit of alone, un-peopley time, to build up my social battery again - so I go and read by myself for a while. But it sounds as if this little girl needs some time after school to decompress from the day, but being expected to be social after school, or to answer questions about ice cream, drinks etc, is the final straw for her - so she needs her mum to recognise this, and to let her just walk home quietly.
I cannot see how this is a bad thing - it teaches the child that it is OK to need a bit of headspace after school. And if she can learn to recognise what is happening, and why, and what she needs to help her cope - that should enable her to cope better, shouldn’t it?
I don’t think anyone is saying that her behaviour is acceptable, but that it is understandable - and with that understanding comes ways that @olafandme can attempt to deflect similar melt-downs at pick up time.
In her place, I would tell her that I can see she is overwhelmed at pick up time, and that we are going to try a different approach, to see if a bit of a hands off method works better for her - so no questions or expectations when she comes out - just a peaceful walk home. If she wants a drink, an ice cream, a snack or an umbrella, she can ask for them, but she won’t be asked.
I would also say that, even though I understand why she has been acting this way at pick up, I was hurt and upset by how unkind she was, and cross at her rudeness, and that, if this behaviour recurs, there will be consequences (loss of screen time, perhaps).