Here's a list of what they tend to do:
- gather information from users on what is/isn't working for them
- gather information from executives about what needs to be done to keep the company working
- gather information from industry leaders who may or may not be part of the company about where the industry is going
- work with design
- take all 4 previous points and distill them into what actually needs to be built for a needed feature
- work with engineering to figure out what is practical, vs not, for building, and how long approximately it will take
- work with engineers / higher ups to remove features that take time but don't add as much value.
- work with engineers to ensure estimation and team velocity is more or less predictable
I've probably already described a 1 and 1/2 jobs. Now if you take product away, each of the above gets distributed among the engineering team, and not everyone in that team is good at it. And even if they get to a good homeostasis, one person leaves and now there's a non-obvious gap that we're not interviewing for.