Do I get it right your problem manifested itself and by itself since you upgraded from UM 22.04 to UM 24.04? If yes, then it casts some doubt whether Mate DE reinstallation may help. Well, if live CD session works as intended, some hope remains still.
Side note: I had an unpleasant experience of crippling my desktop completely. In the end of the day I had to backup my files from home directory and installed the whole OS from scratch.
IMHO, to completely remove Mate DE you need to
apt-get purge --autoremove mate-desktop
As to installation, the following links may be of interest
https://linuxconfig.org/how-to-install-mate-desktop-on-ubuntu-20-04-focal-fossa-linux