If they need to match with the info on paper it's not clear what the case for "digital id" is? If one needs to present "digital id + paper id" one can simply present the paper id as they do today.
That's kinda theoretic discussion by now. As the whole COVID thing is behind us, we can probably look at all the money that were spent in the world to create vaccination certificates, sign them, create the distribution network, distribute the certificates, build the verifying scanners, purchase them en-masse and pay the thousands of people who were standing at the entrance of numerous shopping malls and using these scanners to check the QR codes, only to create a system that is trivially bypassed by using a jpeg file.
My argument is that “digital ids can’t be faked” is a bad argument, and if you rely on it to prove a point then it might be a weak proof.
(Digital IDs indeed can’t be faked but usually they are a part of a process that can be easily bypassed by using something that presents itself as a valid Digital ID even if it’s not.)