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FYI I was able to get to an installation of Catalina. The configuration of /tmp and /private/tmp are the same as on Big Sur and Monterey.

The ls -ald command I recommended should output something similar to this:

drwxrwxrwt  4 root  wheel  128 Sep 15 11:03 /private/tmp
lrwxr-xr-x@ 1 root  admin   11 Sep  7 15:48 /tmp -> private/tmp

If the permissions on /private/tmp do not match what's above (drwxrwxrwt, owner root, group wheel), issue the following to correct them:

# You only need to use the chown command below if 
#   * owner of /private/tmp is not "root" or
#   * group of /private/tmp is not "wheel"
# Otherwise skip the chown command
sudo chown root:wheel /private/tmp
# Use this command in all cases
sudo chmod 1777 /private/tmp

then re-issue

ls -ald /tmp /private/tmp

to make sure the privileges are set correctly.