Tbh I think you're a bit unreasonable.
In your post, it's all about your child and his feelings. He's allowed to cry and scream and tantrum, but the other people round about him aren't allowed to express their (understandable) irritation at his behaviour and how it impacts them.
If it's 'perfectly healthy and natural' for kids to cry and cause disturbance when they are sad, then it must also be 'perfectly natural' for others to express their own upset at this. They should be allowed to express their own feelings too, right? You could say that lady was just expressing her feelings.
In reality, we all have to behave with consideration and respect towards each other in public places. She wasn't considerate, but neither were you. You should have been taking steps to minimise the disturbance your child was causing.
I would not have commented 'shut up' to your child, but if confronted with a screaming child with a parent who was taking no steps to control their behaviour, I might well have made an irritated comment of some sort.
Next time your child has a tantrum in a shop, I suggest that instead of 'validating his feelings', you say to him "Right, you have to stop behaving like this. It's ok to be upset, but it's not ok to carry on like this and disturb everyone else. Do you see anyone else behaving like this? If you carry on, we will be leaving right now."
Maybe see how that works for you?