Anyone who gets a CAC working on a personal computer deals with this all too much. The root certs DoD uses are not part of the public trusted sources that commonly come installed in browsers.
Is this actually all the way technically correct? As far as I know, there is no requirement that the trust chains for server certificates and client certificates are in any way related. It seems to me that it would be perfectly possible for the DoD to use its own entirely private client certificate infrastructure but to still have the server certificate use something resembling an ordinary root certificate.
This is not to say that this would actually be all that worthwhile.
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.
This hits too close to home. I'm sending you my therapist's bill for this month.