Emerson,
There are couple of ways to delete the route stops and the incident in the closest facility layer. If you want to clear out any existing locations before you add something to route stops or closest facility incidents, uncheck the "Append to existing locations" parameter in the Add Locations geoprocessing tool.
If reloding is not part of your model work flow, you can use the Delete Features tool (Data Management toolbox >> Features toolset) to delete all or only the selected features from your stops or incidents.
Hope this helps
Deelesh Mandloi