Unfortunately MATE does not have any built-in colour profile tool.
The Gnome Color Manager was just to check if the problem is with the colour profiles or not.
Now we know it is, but as it seems, Gnome Colour Manager won't help you change it.
This is a recurring problem on MATE forums and unfortunately I'm at the point where I'm not sure I could help any more since I don't have a proven solution for this issue.
Maybe others do have one.
Most of the users recommend DisplayCAL for this purpose, but you have to test it yourself:
Source:
Or an other option if everything else fails is to try the "plain" Ubuntu 20.04, which according to my knowledge, has a built in colour profile manager called 'Color':
https://help.ubuntu.com/stable/ubuntu-help/color-assignprofiles.html.en
I hope I could help, good luck!