You are using a lot of handwaving to try to get across the finish line. Let's tease this apart:
1. "math vs no-math". All engineers learn math. Some are afraid of it. At elite unis, it's fair to say those afraid are weeded out. That does not mean that most who attend mid-tier schools are afraid of math, or don't know math -- that's wildly false.
2. "engineering versus operating". Elite unis prepare you better to go into research, which is not engineering. Yet you are pretending that research is engineering and that what other schools do is teach you how to "operate a milling machine", which is again wildly inaccurate and condescending. Other schools teach you to build stuff and have less emphasis on theory. At elite unis there is more emphasis on theory and less on building stuff.
3. "I've known many engineers who were competent mechanics, but were terrified of math" Yes, as a mathematician I know a lot of engineers who think they know math because they like to throw around words like SDEs, or think they are math heroes for studying a bit of functional analysis or linear programming. But their knowledge is pretty shallow, which is fine. What's not fine is it often comes at the expense of knowing how to build or design things well. That is a real risk with the more theoretically educated engineer, just as not knowing theory well is a real risk for the more practically educated engineer. Also a lot of theoretically trained engineers tend to be quite condescending and arrogant, not realizing how poor their math knowledge actually is.
Fortunately elite and regular engineering schools today do produce good students. The main advantage of the elite school is sorting - smarter students attend these schools, which means on average they are better at their area of study. Caltech is notable for resisting the "equity" tide and still requiring standardized scores, so I'd hire from them if I needed someone to assist in more theoretical areas, or someone looking for a career change into a more practical area. I would be much more circumspect when it comes to UC schools. I would also hire from lower-tiered schools but would require more screening.