Unfortunately cylc set-outputs is the prime offender in the short list of triggering and flow-related commands that aren’t fully user-friendly yet.
For the moment, if you need to use that command, what it does is tell the scheduler to carry on as if the outputs of the target task had been completed. So downstream tasks that depend on those outputs will be spawned, and will be ready to run immediately - unless they have other prerequisites that need to be “set” as well. But you also need to be aware of the flow that the spawned tasks belong to. With cylc trigger the default is to belong to the current flow (and as Oliver pointed out, on restarting a finished workflow there is no current flow). With cylc set-outputs the default is still no-flow, I think, so you’ll need to use (e.g.) the --flow=1 command line argument if you want the newly-triggered tasks to flow on in the graph, even when there is a current active flow.