I've worked as a product manager (and manager of product managers) for over a decade. I do realize that outsourcing the entire product management role is probably a bad idea, but I do believe that there is still an opportunity for consultants to help companies with specific aspects or projects. I want to start consulting and need clarity on the type of problems companies have that a consultant could help with.
#1 would be consumer/product need analysis. Understanding end user needs and developing a product that will generate profit, be adopted or cost reductions is key.
i have seen companies hiring marking research agencies to anticipate consumer / customer reaction to their products, which cost then a fortune... often the loads of data which these agencies deliver in return is also too much to handle, once I saw a consultant recommending a 7 point questionnaire and asking one employee to sit down and call a few hundred customers in a week to get the response. That was much cost effective and pragmatic as well. This might be one thing where consultants can play their part.
Agile training / coaching
Lean training / coaching
Helping the company come up with a development + release process
Remember this: while consultants are a good resource, they must work under a very defined scope of work. The job of most consulting firms is not finish your project as quickly as possible but to make as much as possible off of your project.
This is not to say that consultants are not worth it, just that you need to think through very clearly what you want them to do and in what time frame AND management it.