What does "running the project" mean here?
Agile projects need a Product Owner (and that's usually where projects diverge the most from the ideal - I've never met a PM who actually attended scrums), to stack rank work and answer team questions about requirements. But he doesn't "run the project". It's often handy to have a formal Scrum Master, if you're doing scrum, but he's doesn't "run the project".
Everywhere I've been, management "ran the project", just as with waterfall - they agreed on the delivery date and the staffing. One point of Agile is that the scope falls off, rather than the date slipping, but that's not all that different from tradition.