I'm also not totally clear on the relationship between ecsagent and ecsrun. If I didn't use ecsrun, would the flow run on the ECS service that's running the agent directly?

All agents except local spin up new compute. I think ECSAgent can only pick up ECSRun jobs and ECS agent will never spin a new job on that server. It will make a new ECS Task

So a universal run is basically the same as an ECS run in this case

Yes except without the configurability