I think this is backwards. The public at large are pretty honest that they don't know math. The world would be a much better place if they were able to do practical algebra, even if the higher truths of abstract mathematics never enter their minds.

I really don't see people being confidently wrong about abstract mathematics as an issue. I certainly don't know the rigorous definitions of an integral, but I can apply the concepts of calculus to everyday life perfectly fine. People who care about rigorous math can do rigorous math, and I'm glad if they teach me an intuitive understanding that lets me live a happy, productive life.

Personally, I find layers of abstraction necessary for learning. Maybe there are people who don't, I suspect they would have to be prodigies though. Tell me how to add fractions practically, then teach me the principles when I need to know them. Framing that as a lie seems wrong to me, I'd call it "bounded knowledge".

The confidence is in the ability to learn a topic, not confidence about the knowledge.

It seems to me that the way to build people's confidence in their ability to learn is to allow them to learn. That means not making oversimplified statements because you don't think they're "ready" for more details. It means giving them the details, and letting them decide when they've had enough for now. (Of course one's time, say in a classroom, will be limited, so at some point one has to say "we don't have time to go into this further in class now"--but that's still letting them know that there are more details, and they can dig into them further on their own.)

That doesn’t work for lots of people (including myself). If you’re spending a lot of time on details that don’t actually matter for the topic at hand, it takes away mental bandwidth that should be spent on the topic being covered.