I have created a model using ModelBuilder in ArcMap where points are automatically plotted based on coordinates. I have noticed that the layer cannot be edited and if I try I get the following message; "event later not editable".
This is unrelated to you using ModelBuilder.
As per the Help page on Warning: Event layer not editable:
Event layers that are based on tables without ObjectID fields cannot be edited in ArcGIS.
Solution: Export the event layer to a new feature class or use an editable table for the data source of the event layer.