OK, I'll be the kill-joy here. In the modern STEM world, everyone is a specialist, perhaps over specialised. The idea that there is much value in knowledge of disparate engineering disciplines is quaint, more in the realm of winning Jeopardy than actually useful. As someone who had a long successful career as an EE, I can assure you that I never needed to know how to calculate the strength requirements of a bicycle chain ring, or what alloy it should be made of. Same with Navier-Stokes, refractory columns, upper-state lifetimes of YAG vs. YLF, airfoils, etc. I can also assure you that a chemical engineer isn't paid to design electronic circuitry for real world applications.
The guy that works on the ECM in a Camry doesn't need to know how to design the shock absorbers and vice-versa. They do need to choose early in their education which kind of engineer they want to be, and they need to know how to work with all of the other engineers. Engineering is a team sport; pitchers don't take batting practice, and catchers don't play in the outfield.
"The beginning teacher... Applies mathematical principles of manufacturing processes in lathe operations and computer numerical control mill programming to model and solve problems."
--- Either that's your career, and you'll learn everything about it, or you'll hire a machinist to do that for you (note: a machinist, not an engineer; clearly this wasn't written by an engineer).
I think anything beyond a general survey class, where students learn that Chemical Engineering exists, is probably a waste of time. It is the STEM core; math, physics, maybe chemistry, maybe biology, that ties them together. i.e. general knowledge, so that when you find out that you do need to learn a little bit about the other guy's job you have some of the fundamental tools.