Right. Airlines will ask you to show your card when you check in for a flight, just like they ask you to show a visa (to countries that require it).
No one is going to have the airlines integrated into the system, not for a long time.
You know that company Clear which does all the TSA fast-lane screening when you go to the airport -- the screening for COVID vaccinations will merely be integrated into this. Airlines will need to do nothing more than ask you for your Clear voucher with the QR code that you got from the Kiosk validating your status before your international flight. The airline will scan the QR code on your Clear voucher (in a Clear Vaccine Passport reader) when they look at your passport before you board the plane. There will also be an electronic version offered by Clear on your Smart phone with the QR code so you don't need the paper voucher from an airport kiosk.
And yes... they have this all prepared, tested, and ready for deployment & use.