Follow up:
the patch update failed because the root password (or the admin password) needed to be updated.
You can see this in /storage/log/vmware/applmgmt/software-packages.log
So I updated the root and the admin passwords, installed the update via ISO, and now I am on version 6.5.0.22000 Build Number 9451637.
BUT:
VAMI was still not up after a reboot. All services looked OK in the VCSA console, but vami-lighttp was disabled in the OS. To start it, 2 options:
1) go to the console/shell and run "service vami-lighttp start", every time you restart VCSA
2) go to the console/shell and run "chkconfig vami-lighttp on", and reboot. (once)
I marked this as the correct answer as it is more complete, but I would have been unable to get here without the help from a.p.