I worked in telecommunications for 27 years so have always been up to speed with this since the late 1970s. I now work freelance in another field and 100% of my work arrives via my mobile phone - as voice calls. I have a Smartphone for the usual reasons. However, when I travel and have a boarding pass in the relevant 'app' of the carrier, I still print out the pass and carry the paper. I have printed confirmation of every flight, car and hotel booking in a trip. At the end of the trip, they go in the recycle bin.
I can see that, in the early months, one might have to register vaccinations with your carrier AND another scheme. For those not regularly using modern technology this will be limiting. We might hope that the carriers will realise this early on and establish a 'vaccine check-in desk' to stop at before bag drop. It will be cheaper than senior citizens not travelling.