Pizza sized teams that can directly interact with customers don't need management. Management is playing telephone with someone in the middle with their own interpretation of everything which is 99% wrong.
Those don't really seem like things you'd want to outsource, no matter how good the outsourcee is.
Product people are not management at least not where I worked.
I see how it might be a problem if your manager is talking to the customers.
Product people are or should be equal to dev, engineering. They should indicate priorities but should not in any way evaluate engineers especially if they have no technical expertise.