Cannot copy directory in MacOS with terminal Input
$ pwd /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave Information

$cp /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave\ Information /Volumes/old/QT_screen_recs

Output
/Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave Information is a directory (not copied).

I don't understand why this is not working.  I've also tried:
$cp /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave\ Information/* /Volumes/old/QT_screen_recs

Which yields:
/Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave Information/*: No such file or directory
When you want copy a directory with command cp you need add option -r or -a.
-a is best because it preserves the structure and attributes of files as indicated in the man command :
man cp

In your case if QT_screen_recs is a directory the corrects commands are :
cp -r /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave\ Information /Volumes/old/QT_screen_recs

or
cp -a /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave\ Information /Volumes/old/QT_screen_recs