I think you're right that it's possible in principle for a Web server to enforce use of DoD CAC (enforcing the client cert being in the DoD PKI) without itself using a DoD PKI cert on the server side.
That said there's little benefit to it, users who haven't jumped through hoops to install DoD root CA certs won't typically be able to get their browsers to present them to the remote server in the first place, and if we're willing to jump through those hoops then there's no good reason for the DoD server not to have a DoD PKI cert.
FWIW I just tested it and yes you can run a web server using a commercial server cert that enforces client PKI tied to the client having a DoD PKI cert. It works just fine.