I think you don't understand the embarrassment because you're trying to pick this apart too surgically. First of all, I should call out that I think our conversation is really about a hypothetical situation in which a display like this was operating in an airport for all passengers, whether they opted in or not.
That being said, I think it would be totally fair for a person to feel embarrassed about any personal information being displayed publicly on a monitor without their consent. Sure, all of this information could in theory be gleaned from other sources, but I don't think people are bothered by this because they know it would take some particular effort by someone who was interested. That falls under the average threshold of concern. What's more, it's just a fact of life. I don't think people expect water tight personal information security. But I do think people have an expectation of privacy. The example you gave is illustrative. You said someone can ask you for your first name. If they're an airline employee, fine. That's necessary for them to run their business properly. But if they're a stranger on the street, you're perfectly entitled to tell them to get lost. Now let's consider that a hypothetical, non-opt-in PI display in an airport would be rather like a stranger walking up and asking you for your first name. Yes, there's not much actual threat in giving it. But who would want to do that? No one. I'm not sure I could say why, but I don't think it's fair to ask why and I don't disagree with the instinct not to hand out one's first name on demand.
I think it's important to take into account these kinds of "first-person" concerns that have a sort of common sense to them. You might be able to diffuse them from a strictly logical point of view. But that doesn't do away with the fact that an average person would still have the concern. And maybe there's an unseen logic in the common sense viewpoint.