If an internal page is outranking the homepage that is buried deep in the serps, I would look to see if there are any potential penalty issues. It is possible that having the "the most backlinks by far" might have triggered a link penalty. Even if you have several .gov or .edu links, there might be some poisonous links as well that will sink a good website.
Another possible reason they are having trouble is that spending alot of money and effort in the wrong direction will not help you rank. I still come across companies that spend thousands of dollars on meta keywords optimization which stopped helping Google rankings years ago. Big companies with big budgets aren't perfect and sometimes they make big mistakes.