I don't share your embarrassment. There's a significant chunk of the country that doesn't understand the issue - they think the claim is about blatantly discriminatory practices. In some cases, I'm not sure that even the people who do advocate for solutions to the pay gap truly understand what the core issue is. Both of this caused by the way all of this has been worded.
Worth clarifying all of this as much as possible to work towards putting everyone on the same understanding of the situation.
Yes, the wage gap isn't literally about cigar-smoking men in dark rooms deciding to systematically pay women less. It's more like someone who claims not to see gender paying women less because men negotiated or "seemed more confident".
I'm embarrassed because this kind of stuff can and will be used to continue the narrative of "women make just as much as men" which isn't true.
Worth clarifying all of this as much as possible to work towards putting everyone on the same understanding of the situation.