Does carrying just a QR code work fine for you? My experience from Germany is that nobody scans the code, they just scroll to details to see there's two vaccinations recorded. This far my QR code has only actually been scanned at the airport.
There is however a prevalent issue with QR code scanning that I noticed twice personally (and friends reported the same experiences at other places), which is that most restaurants use a super cheap phone/tablet with a terrible camera that usually is unable to focus on a smartphone screen correctly.
Once, the person was honest about it and explained they kept having issues with it, and I tried to help him focus correctly his device (QR code size + luminosity seems to be the issue from what I saw).
The second one tried a couple of time, didn't manage to focus and told me "all ok" despite it's device not having scanned me (I could see the screen).
If I had to travel though, I'd definitely take the paper certificate on top of it, as, at least here in France, that's considered for better of for worse the source of truth.
1. Scan the code with the app every time.
2. Ask if I have the pass, maybe look at it once, hopefully remember me the next times.
There is also a small percentage of cases of no questions asked, ever.
By the way, I'm using a printed copy of the pass. It never runs out of battery, I don't mind if it falls on the floor when I'm using it, it works even if I'm using the phone for something else. Cons: it needs to be replaced if its key gets revoked.
In Ireland it’s hit and miss. About 60% of the time had to produce it, but very few places want to see ID as well, which is needed to prevent people just using someone else’s pass.
I’m all for it so I’d love to see better enforcement.