It's definitely a solution when you have an app (assuming you have a 3rd party source of identification of users), but since it's a French key, a sizeable portion of people, mostly elderly, are not using the "official" app but a paper printed QR code that was given to them at the time of the second injection (or a picture of said code).

Since the French app is pretty terrible (it used to be the barely functional contact tracing app that didn't want to use Google/Apple's API, so there's a pretty strong popular stigma against it), even a sizeable portion of smartphone users use an alternative too (I personally have my QR code in Apple's Wallet, but there are other 3rd party solutions).

Depending on when that key was in use, this may be an epic bureaucratic mess to solve here in France if they have to reissue many.

Edit : I'll quickly add that while there's a database of vaccinated people handled by CNAM (the single payer social security administration), it doesn't seem extremely accurate : my best friend and one of his collegues were vaccinated, issued a QR code, but they weren't added to said database and had a terrible time getting the situation fixed. In the end, a pharmacist reissued a "vaccination" for them to "get them in the system". So that database is probably at best incomplete (for the process of reissuing).