When selected via photoshop or other software the ICC profile takes no effect on the print, and the printer seems to receive raw unconverted RGB data.
Then something may be wrong with your install of Photoshop. Use the Adobe supplied uninstaller in the Utilities folder, then reinstall Photoshop from Adobe's server via the Creative Cloud app. Test again.
I’ve looked around online and it seems this has been an issue for some folks since Big Sur.
Certainly not impossible, but I've had no issue using profiles in any version of macOS, or with any app.
Perhaps I'm misunderstanding you--do I need to have the ICC paper profiles installed in /Library/ColorSync/Profiles for Photoshop to actually make use of them?
No, they can be in any of the three normal locations. One of which used to be created for the user, but now needs to be added yourself.
/System/Library/ColorSync/Profiles
/Library/ColorSync/Profiles
~/Library/ColorSync/Profiles
The first, being in the System folder, is completely untouchable. Ignore it. The second is the normal location where all user accounts can see and use all installed profiles. The last you now have to manually create by going to your user account Library folder, then creating the folder ColorSync, and then a Profiles folder within that. It serves no purpose to do the last one unless, for whatever reason, you don't want other user accounts to see those particular profiles.
Just to clarify, are you using these settings (other than the selected profile show here)?

For images, always use Relative Colorimetric for the intent. The rest are garbage. Always have Black Point Compensation on.
Per MartinR's notes, printer profiles are really paper profiles. Such as, if you have an Epson printer and choose an Epson matte paper profile, that profile is only useful for the printer model it was created for, and they truly mean Epson matte paper.
If you don't have the same model printer the profile was made for, you won't get usable results. If you use HP matte paper, you won't get usable results. It doesn't matter that the Epson and HP papers look and feel the same, they will not react to the inks the same way, and the color will look entirely different on the two types of paper, despite using the exact same output settings.