You sound like a jaded professional, but keep in mind most people are not working mathematicans and never will be, so anything they hear from the pros they're g

You sound like a jaded professional, but keep in mind most people are not working mathematicans and never will be, so anything they hear from the pros they're going to take at face value. When it comes to the general layperson, I think popularizing the idea that math is all "months of flailing around, trying to understand one thing" is more unhelpful than portraying it as the pursuit of beauty and simplicity, even if it's more true on a practical level. If you bring up the word beauty in reference to math, most people are a little surprised you even would link the two concepts, let alone get the impression that pursuing clever little proofs is all there is to it. Better to start by giving people get the idea they might enjoy it, and let them find out it's not all about elegant proofs in undergrad or whatever.