Hello Dan,
In a "domain" environment, computers can be configured (via a GPO or otherwise) to auto-enroll for certificates. This is probably the "normal" case for most companies.
One can direct individual computers to request a certificate from a certificate authority - the computers can then generate their own private keys (no need to distribute these in .pfx/.p12 files).
Another option is to distribute certificates and keys manually, as you are currently doing; the long-term manageability of this approach is questionable.
When manually importing the .p12 file, it would be best to use the "automatically select the certificate store" option, since normally the computer specific certificate should be stored in its "personal" store rather than the Trusted Root Certification Authorities store.
Certificates expire and will need to be replaced (perhaps automatically); rather than depending on identifying the particular certificate to use, one specifies the issuing authority (CA) - the client then selects a valid certificate from its personal store issued by that authority and that matches the purpose/use (e.g. client authentication).
Gary