I agree with Stanislov in general.

Specifically Product Management as a Service in my case includes:
1. Due diligence for products and experiences
2. Recruiting or mentoring PMs
3. Tools and methods streamlining and implementations
4. Helping with strategy, product-market fit, market research etc.
5. Taking features end-to-end, from requirements till launch

#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.