when constraints compete, the factor that drives the decision is the #1 priority
That's simply wrong, and apparently contradicted by your last sentence about
balancing cost, time and safety. Let's suppose profit is my #1 priority with safety at #2. If I could double my safety by reducing my profit by £1, should I refuse to do so? No, of course I would take a balanced view and decide that the tiny loss of profit could be outweighed by the large improvement in safety. But that wouldn't alter the fact that profit remained my #1 priority.
And FYI, I'm never impressed by the "I'm a management consultant, don't you know how important I am?" line. I've met quite a few able and competent folks who called themselves management consultants - and a lot of crap ones, too.