You have to pick metrics to show management that they understand the importance of, and then be able to demonstrate degradation in those metrics when you don't

You have to pick metrics to show management that they understand the importance of, and then be able to demonstrate degradation in those metrics when you don't do what you propose, and improvements when you do. Accomplish this and you'll build trust. Too often, what happens is that the proposed benefit is vague and not empirical, and then the benefit is not actually realized by a large investment into it, destroying trust.