In other areas of my life, having a sense of urgency helps me get things done in a timely manner - like doing grocery shopping or running errands.
I think that urgency artificially introduces stress to the task - which may be beneficial in certain scenarios.
Knowing (with fair warning) that something is going to be late lets me set the right expectations. With a month to go I can usually persuade a customer to slip a week because something just needs needs more time. I can also reconsider my time estimates for future jobs to make sure that in future the engineers get the time they need.
What I can't do is keep promising delivery "tomorrow" to external people, because I just found out today that you won't be done on time. That's bad for everyone.
It works great for all involved: I get to do quality work I'm happy with, and they always understand where their time goes. I admit it doesn't work with all PMs: some will try to mandate a due date. I'm really lucky to have a boss who'll help me push back, to explain that wanting things more doesn't make them go much faster. And I'm doubly lucky to have a boss who's right when he (very infrequently) has me work overtime to get something that's actually urgent done.
I find stress much less motivational than the pride in doing things well.