What arcpy function deletes ALL database connections in ArcCatalog?
There is no arcpy function to delete all database connections.
A database connection is simply a file on disk. It can live in any folder, but if you're in ArcCatalog and talking about multiple connections you're probably referring to the Database Connections node. That is a virtual path in ArcCatalog to an actual folder on disk.
If you want to remove these files you'll need to identify that directory in your profile and write your own code that has nothing to do with arcpy. Do a quick google search for
python delete files in directory: the first answer is a helpful stack post.