The thing is, this type of record isn't designed to help first responders. As you pointed out, you show up and try to figure out what's going on.
It is a lot more useful in other situations. Let's say my usual GP surgery is closed, I could just walk into any other, have an equivalent of France's "Carte Vitale" scanned, and the GP has immediate access to my records, including prescription history, so they can re-issue a prescription for example. No need for registration. There are "pharmacist precribers" as well, so if they have your records they could e.g. renew your prescription for insulin, without you wasting doctor's time for what is effectively an administrative function. After (limited) renewals, you would have to go see your doctor again, but that could be customized to the drug/patient, or over-ridden entirely.
Again following the French system, once I have my prescription I got to the pharmacy get my drugs. The pharmacist scans my card, and discounts/waives are applied automatically.
Also it prevents "double prescribing". A distant relative of mine had studied which symptoms to describe to get which drugs she wanted, so she had a GP prescribing her antidepressant, another sleeping pills, another heart drugs, etc., and picked it all up from different pharmacies so they wouldn't pick up on interactions. All for free (well, paid by French taxes anyway) because of special circumstances. She had to give it up when the Carte Vitale came into place. Had to go cold turkey, but did her good in the end.