Any ICC files installed by the operating system are kept in /System/Library/ColorSync/Profiles. As this is now on a read-only System partition in Catalina and Big Sur, applications that attempt to install ICC profiles there now will not succeed, even with admin privileges.
There is another /Library/ColorSync/Profiles location that may receive ICC entries with your admin privileges, and where newer applications that install ICC profiles should now be installing them, and applications that expect to find ICC profiles also now look.
And thirdly, it may not exist yet, but you can create a new ColorSync folder in your local Library (/Users/username/Library) with a Profiles folder within the ColorSync folder. You can deposit ICC profiles in that Profiles folder without any admin privilege.
What is not known is where your unnamed applications that use ICC printing profiles are looking for them when attempting to print, and whether this location is tunable in their application preferences.
The remaining question is whether ICC profiles become outdated, and newer applications refuse to use these outdated profiles, preferring newer versions.