Please try this command to fully remove the desktop
sudo apt-get remove --purge ubuntu-mate-core ubuntu-mate-desktop
If you have both GDM and lightDM installed please
sudo apt-get remove --purge gdm
Then set light dm to default (use the method you said you use before)
Based on mchids comment below this, you need to edit edit this file 50-unity-greeter.conf
Create a 50-unity-greeter.conf
sudo gedit /usr/share/lightdm/lightdm.conf.d/50-unity-greeter.conf
The file this opens should be blank so add this to it
[SeatDefaults]
greeter-session=unity-greeter
Save it then
sudo chmod +x /etc/lightdm/lightdm.conf.d/50-unity-greeter.conf
Then reboot and see if this helps
TO FIX YOUR ICONS
An easy fix for your icons should be install unity tweaktool from software center (dont know package name for apt, maby someone will edit)
Go to icons and just choose different ones
Best Answer
To completely remove MATE:
Uninstall mate-desktop
To remove just mate-desktop package itself from Ubuntu 16.04 (Xenial Xerus) execute on terminal:
Uninstall mate-desktop and it's dependent packages
To remove the mate-desktop package and any other dependant package which are no longer needed from Ubuntu Xenial.
Purging mate-desktop
If you also want to delete configuration and/or data files of mate-desktop from Ubuntu Xenial then this will work:
To delete configuration and/or data files of mate-desktop and it's dependencies from Ubuntu Xenial then execute:
REF:LINK