It might be like that. First, anyone approving a budget knows what they're told, but probably has the capacity for very little first-hand verification. Second, not all budgets are "given" - we've previously seen the CIA raise money independently to fund illegal operations.

Let's put it this way: the president, and the congress could squash that in a second if there was a will.

Plausibly. I'm not sure how we'd tell. And there being "a will" may be dependent on an understanding of what's going on whose absence could - in principle - persist indefinitely.