Crossing the border based on tests means you're paying the cost of the tests and going through the aggravation and time of getting it done. Vaccine administered in your country of residence is free of charge. That alone will drive people who travel towards being vaccinated. But then comes the matter of proving that it's done.
Vaccine passports for international travel, whether it's called that or not, are going to happen. Yeah, perhaps testing will be accepted as an alternative, but it will be more aggravation and time to do it that way.
"vaccinated people can still carry and shed the virus" ... and people can still acquire infection in the non-zero time period between submitting a test sample and arriving at a border crossing. There is non-zero risk either way, but mounting evidence is that the risk of being infected post-vaccination (more than 2 or 3 weeks after) and getting sick enough to be infectious, is very very low.