Engineering school is almost entirely math. It's applied to structures, to electrical circuits, to fluid flow, but the essence of it is all the same - using math to describe things.
Yes, a lot of design doesn't require calculations to be done on the spot, but almost every design made by an engineer will be informed from the beginning by their knowledge of math.
It's hard to think of a modern invention that's as big as the internal combustion engine or the aeroplane, but any recently designed item you buy today has had math applied to almost every facet of it. The design process for a product case uses extensive geometry, the design of gearboxes uses extensive mathematical modelling - pretty much every mechanism that's manufactured will be first described mathematically.
Even things that don't have complicated mathematical models describing them will have had some form of math applied to inform the initial design process. You can't design a lever without taking into account it's length, the moment it applies etc - that's all maths!
Technical greatness in our world requires math. You can't design and build a truly unique and functional device without applying some form of math to it.
Certainly spacial awareness is absolutely fantastic. It lets a designer envisage a concept in their mind, think through how something will work and identify the pitfalls of a concept before anything is put to paper, but it's not a replacement for mathematical ability when it comes to actually designing a functional mechanism or mechanical system.