The opinion above is that proving you are vaccinated may violate the law. Time will tell on that both for workplaces and transportation. I see no reason that airlines cannot deny service for lack of vaccination. A negative test may be a work around for those who are religulous.
And that is why/how the US airlines will likely operate. They will likely not be able to ban you from flying without having been vaccinated, but they can require require a negative Covid test to fly...................which is exactly what they do now.
Nothing new there.