Just take a look at JCE Editor how this is done....
Just before any actions in SP Media Manager, press Browse Folders and you will get access to the same structure as in JCE.
The window in which you see the More button is designed for quickly access to the last uploaded files. It is very convenient and do not need to change anything there. In this aspect, SP Media Manager has an additional option that is not in JCE.
The only thing that I suggest for developers to make is to replace in this window the Upload Files button link to the folder structure instead of the file uploading action. It is this button that is misleading and uploads files to the wrong folders based on the date.
You just need to get used to and understand the logic of the work. After that you make sure that everything is fine.