One of my mantras to my associates and consultants was that our job was to be one part therapist, one part sociologist, and one part accountant. As you say, if the client could fix their own problem, they wouldn't have called in a consultant, and, if the problem could be fixed by throwing developers and designers at it, they would have called someone cheaper. The implication of getting that phone call is that there's a gap between what the client's
asking for and what the client
needs, which has led them to bring in an outsider. Generally the answer to what the client assumed was a technical or design issue actually lay somewhere in the organizational structure, interpersonal relations, or the cash flow and balance sheets.
It's true that consultants generally (certain kinds of expertise-led consulting aside) don't do anything an organization can't do for themselves, but it's also true that a therapist doesn't do anything an individual or couple can't do themselves. Bringing somebody in to ask both the stupid questions and the hard questions, without being beholden to internal power and communication structures, can often be very helpful, as long as -- big "if" here -- you can trust the consultant isn't going to featherbed for the sake of the SoW.
Having said all that, I've seen some terrible, terrible work from consultancies, including some big screwups from my old firm (which, to their credit, the cost of which we ate rather than our clients). More than once I've seen large clients outsource strategy, innovation, or the entirety of execution to a Big Three/Big Four firm, and waste tens of millions on work that delivered a couple dozen binders but not any tangible value. The common thread is usually that the company abdicates responsibility for overseeing and critiquing the work, rather than taking a firm hand. That never turns out well.