I don't think MyMaps will do what you're looking for.
The "Export to KML/KMZ" dialog you are showing is for exporting from MyMaps, and if you select "keep data up to date...", it will give you a KML file containing a NetworkLink to the KML output of your MyMap. If you view that KML file in another KML viewer such as Google Earth, any edits you make in the MyMap will show up in Google Earth.
Unfortunately it doesn't work the other way around. If you take the KML file you get from MyMaps and edit it, there's no way for those changes to go back to your original MyMap. Even if you upload the KML to Google Drive, the MyMap won't auto-update. MyMaps import tool is a one-time shot. So the only way to update the MyMap would be to re-import a KML with new/updated date.