Now they might not be running their own infrastructure, but they still have to be the one who authorized the credential to be minted. A faux digital National ID card issued from a random social network will likely not let you get access to government services or let you open a bank account, even if the data itself is all correct.

It is more likely that you'll have "trust frameworks" describing technical and logistical interoperability, and verifiers will accept any issuer that falls under that umbrella. In the US with driving licenses and state identification cards for example, that would likely be AAMVA.