"In the backend, it has a small SQL database which stores no personal information: just the list of invitees and their status." This status is private medical information : in France (and all Europe probably) it is illegal to store these datas, unless you use specialized "health datas secure hosting". You cannot legally do it yourself this way. French Official Documents : Health Datas Hosting : https://www.legifrance…
Storing name, DOB, test date, test result in the QR code is a good solution IMHO and indeed using a database as a private organisation is fraught with problems. Name and DOB are reasonably needed to assert the person's identity. Especially in France everyone carries an ID card so people can show ID and certificate together. Safeguarding private information is important but some must be disclosed as necessary.
I think this is only the address one gives freely to the vaccination/test center and which is not checked by anyone.
I still can't think of any case where it would be useful in conjunction with your vaccination status.