Being embarrassed is a moment, an emotion.
Losing face is like being embarrassed in a small town. It's like a more durable, persistent embarrassment.
Everyone feels this way. Everyone needs a term for it.
Chinese people are just people. Some are totally shameless, just like everywhere else. Most are normal and feel the same things you do – just like everywhere else. Let's ease up on pretending China has a deeply strange culture because of this.
Let's pretend they have a deeply strange culture because of their sense of humor, which is truly quite different.