The "Default" profile is loaded first. Since you've previously played videos using the "Default" profile, those videos appear in your media library. You can clear those files from the media library. Click on the text that says "Media Library", and then click the delete icon next to each entry. The media library shouldn't show the files from your private profile if the "remember last profile" toggle is disabled.
If you're referring to the player showing clickable files/folders in the file explorer, then there isn't a way to prevent that. The player remembers the last folder that was opened for each profile, so you could make sure the last folder used with the "Default" profile is one that you're comfortable with others viewing.