And hey, maybe before college, almost everyone is a non-mathematician. There's the small majority of students who'd love learning group theory because it's fun and beautiful, and then there's everyone else who need practical applications and real-world examples. But the way you teach them is ultimately just like the way you'd teach non-mathematicians in college or further on in life, when there's a clearer delineation.
It was a success overall for the Engineers but Math wasn't happy. The engineers working on advanced coursework needed higher level math courses that were only available in Math. Not only were they constantly failing which angered Engineering but the professors teaching them had to devote more time to the Engineering students which took away from the math students and angered Math.
The sort of uneasy truce that they eventually came to was Engineering students take the regular Math courses and their Physics/Engineering curriculum supplements what they're taught in Math. This annoys double major Math/Eng students because there's far too much repeated material in their curriculum but it's the best they could do.
I did not find anything really surprising in the first part of Gowers's essay, but I thought the list of questions was great, independent of whether or not it is really appropriate for a required course.