I think we solved part of the issue.
I wasn't aware that there is a Library folder in my user folder, because it's actually hidden (as it's supposed to, I've learned from a Google search).
I managed to access it, and indeed there "J River/Media Center 32" was present, meaning that the remove command probably didn't delete it (which is weird in itself).
After having deleted the Media Center folder manually and reinstalling MC 32, finally the Media Center window popped up asking me for my license.
However, after restoring my license, the program instantly crashed again.
I replicated the process (removing Media Center 32 from ~Library/Application Support/J River and afterwards from the bin, reinstalled MC 32, restored the license) and it still crashes (crash report attached).
Again, permissions are all in order, antivirus is disabled, and connections are allowed through Firewall.