I figured it out before I finished writing this question so I figured I'd just share this with anyone else.
# ls /etc/motd.d -lh
total 0
lrwxrwxrwx. 1 root root 17 Mar 12 19:46 cockpit -> /run/cockpit/motd
lrwxrwxrwx. 1 root root 41 Jun 18 22:01 insights-client -> /etc/insights-client/insights-client.motd
Each one of those symlinks prints something to the SSH terminal on login. In this case, cockpit
prints the web console message and insights-client
prints the insights prompt.
Therefore just do this:
# cd /etc/motd.d
# [sudo] rm -f cockpit insights-client
Verified to work and persist across restart on RHEL8.4
.