It’s a social situation where someone getting shut down is a humiliation

Sure. I'm just saying that, for the typical person, the humiliation from others being present is probably very little compared to the humiliation from the actual rejection. Maybe that's different at college bars or something, but in most regular bars, most people don't care. At least for me, I'm just as nervous walking up to someone in a bar as I am at a club (even fewer people looking) or something, approaching a stranger is the main thing causing the nervousness.