I'll get made to take shortcuts to get the product out before I'll be allowed to be strict on correctness if it means pushing a deadline back even a single day. I love correctness, but it's just not often important to people who aren't me and for reasons completely outside of my control.

Correctness is important to everyone. Presumably everyone involved would rather software work rather than not work. Time commitments are what they are and sometimes quick and dirty is what you have to do but I don't actually think there is some hard tradeoff between fast and correct. The whole idea of having rigorous foundations is that you have a set of primitives which can compose together in well-understood ways so you can build correct programs quickly without necessarily re-inventing the wheel each time.