Config sync directory outside of webroot causes open_basedir restriction warning
PHP should have no problem with seeing outside of and jumping out of the webroot, correct?


Very much not! As the errors are telling you, your server config explicitly doesn't allow that to happen. You have PHP locked down to only be able to read from certain paths; /home/cathchar/config and/or /home/cathchar/config/sync don't belong to that list, so PHP can't read files from them.


  Do I have to inform Drupal somewhere else about a new path to work in?


No...Drupal doesn't need to be told about file system restrictions, it just consumes the environment it finds itself in, and invariably complains if it can't access what it's (implicitly) been told it can have access to. 

This should ideally be solved in your vhost/FPM config by setting an open_basedir which includes all of the paths that PHP needs to be able to read from (separated by a ;).


  None of the tutorials/instructions I've read ever mention issues regarding directory access, paths, or open_basedir restrictions


The docs don't really try to cover every possible configuration scenario, it's probably assumed that if you're doing something more advanced like moving a folder out of the web root that you're happy configuring the rest of the server to allow it to happen.