As recommended by fossfreedom, just install the relevant meta package, with or without snaps. I did that and it worked out well.
But, same as you, I have package conflicts with Mate ever since.
I removed every Mate dependency (Marco, Mate, etc…) by picking them exhaustively (checking for all the Mate related terms in every package name, description, version or dependency) and yet in Synaptic or Ubuntu Software updater (Gnome) if I let updates go normally it re-installs everything Mate. Everything that is uninstalled and (seemingly) free of dependencies.
Ever since, I have to go into Synaptic, Status → Installed (Upgradable), select all, mark for upgrade and apply. I must avoid the button at the top to “Mark All Upgrades” or it will reinstall everything Mate. I looked it up but couldn’t figure it out. So I suspect Nemo, as it’s the only thing that seems related.
But I don’t have that issue on my laptop, although nemo is my go-to as well and Budgie is installed from Ubuntu (not as Ubuntu Budgie). So in the end, I’m not 100% sure if Mate is responsible. And I couldn’t find the dependency that require everything Mate.