Another thing to "worry" about, sometimes while the file is "closing" ... it might be impossible to modify anything inside the file. That's usually why the "command will start" reactor is usually better to use. If you use a :vlr-beginClose reactor, it might fail, especially if the user clicks Cancel when asked to save changes. Also you might need to undo the save on a :vlr-commandCanceled / :vlr-commandFailed reactor.
Reactors, dangerous they are ... you find all sorts of "disturbances in the force" which you never thought would happen.
I also think it might be a better idea to have the log saved as a text file in the same folder as the DWG with the same filename. That way it might alleviate any non-modifyable issues. Perhaps save it in CSV format, then you could link to it as if it's an Excel sheet and have a auto-updating table if you really want it displayed inside the DWG. Then you might even use the :vlr-databaseToBeDestroyed since you're not going to modify the DWG database at all.