Yes, the story is an urban legend, the budget goes up every time it's told, etc. But I think the important lesson here is not really "line worker outsmarts expensive consultants", its really about
incentives. The consultants are incentivized to do good work (they want to keep a good reputation and potentially get more work) BUT they are also incentivized to charge as much as possible ("value-based pricing" is the term the industry loves) which means they will often try to convince you your problem is more complicated than it actually is
Similarly, the line worker is likely (correctly) to believe he won't be paid much more if he works harder, so his primary incentive is to just reduce his overall workload as much as possible and still get paid.
The best solution (and I believe a bunch of research backs this up) is to better incentivize your line workers and reward them appropriately. I know one study in particular that showed broad-based stock option ownership among employees was correlated with stronger company results, while options concentrated just among senior execs was not correlated with business success.