That’s normal and I’d say (gasp) acceptable. Even in bigger companies, maybe even in enterprise-level companies. Completely depends on how the division/department/unit is structured and tasked.
In the organization I work for, we have 120 people in the marketing just in HQ location – a dedicated editorial team, a dedicated design team, a dedicated multimedia team etc and yet almost every single piece of content that is published on social media platforms are created by the social media team.
In my experience, social media management is most of the time a content + community management game. If you are looking for a unit head or strategist position, look for “social media lead”, “digital strategist” and similar – although planning around just the social media will not be enough for the latter one.