Shushing a child in public communicates that you care more about how everyone else around them feels than how they feel.
Think about it for a moment. A child begins to fuss, whimper and even cry or scream. And what does the parent do... Do they first ask themselves about what the child is feeling, thinking, having trouble with? Do they look around pleadingly or embarrassed? Do they scoop the child up and rush off? Do they chastise the child? Do they tell the child how to feel, or how not to feel? Do they encourage the child to understand how they are feeling, what they are responding to? Do they yell? Are they calm? Do they bark commands? Are they narrating and showing the child their own thought process?
I think it is always important to first acknowledge their emotions, then grant them the awareness of their surroundings. It is important to recognize how they feel, to give that the first priority. It's also important to teach them about social norms, rules and expectations. But it's how we do so that builds an enduring trust and bond.
For example, saying something like, "You look really upset, because you want ice cream, but we're not getting ice cream today. You really really wanted ice cream. I wish we could give you ice cream every day, whenever you want. It's really upsetting not to get what we want when we want it. There are also a lot of people around, and yelling is not for the supermarket because it is disruptive. We use indoor voices at the supermarket, because other people also use indoor voices and everyone needs to do their shopping. When we are upset, we can say 'I am very upset because I want ice cream and I can't get ice cream' instead."
While it may seem long or convoluted, it communicates a few very important things. Firstly, that we are attuned to the child, their wants, needs and frustrations. That those things are very important to them, and to us. That we care about them and want to make them happy. That they won't always feel happy and we won't always be able to give them what they want, regardless of how we and/or they feel. That we need to be conscientious of our surroundings. That there is an appropriate way to communicate their feelings. (It is also to recognize that we need to be attuned to the way a child
feels, because if we aren't they will have to escalate in order to
ensure we're getting the message.)
If the child is still very upset, then they probably did not feel as
though their feelings were truly heard and understood (some kids may learn that they are really only heard/understood when the parent responds with action or gives them what they want). So it's important
to focus on their feelings. Sometimes, it's even important to leave (especially in
a public place), though that can be an additional frustration for the child.
Otherwise, by shushing them, we just communicate to them that we care less about their feelings, what they are experiencing and who they are and more about how other people they think of us. Or maybe, we communicate that they need to try harder to really get our attention and make them a priority. Make no mistake, they will try harder, push and escalate. Children, more than almost anything else, want to be heard, understood and responded to. The attention and empathy is so important to them, it cannot be overstressed.
We don't always have to do what the child wants, but it's often much easier on both the child and the parents when we give them our understanding empathy.